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<a name="math_toolkit.stat_tut.weg.binom_eg"></a><a class="link" href="binom_eg.html" title="Binomial Distribution Examples">Binomial Distribution
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<div class="toc"><dl class="toc">
<dt><span class="section"><a href="binom_eg/binomial_coinflip_example.html">Binomial
Coin-Flipping Example</a></span></dt>
<dt><span class="section"><a href="binom_eg/binomial_quiz_example.html">Binomial
Quiz Example</a></span></dt>
<dt><span class="section"><a href="binom_eg/binom_conf.html">Calculating
Confidence Limits on the Frequency of Occurrence for a Binomial Distribution</a></span></dt>
<dt><span class="section"><a href="binom_eg/binom_size_eg.html">Estimating
Sample Sizes for a Binomial Distribution.</a></span></dt>
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<p>
See also the reference documentation for the <a class="link" href="../../dist_ref/dists/binomial_dist.html" title="Binomial Distribution">Binomial
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<a name="math_toolkit.stat_tut.weg.binom_eg.binom_conf"></a><a class="link" href="binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution">Calculating
Confidence Limits on the Frequency of Occurrence for a Binomial Distribution</a>
</h5></div></div></div>
<p>
Imagine you have a process that follows a binomial distribution: for
each trial conducted, an event either occurs or does it does not, referred
to as "successes" and "failures". If, by experiment,
you want to measure the frequency with which successes occur, the best
estimate is given simply by <span class="emphasis"><em>k</em></span> / <span class="emphasis"><em>N</em></span>,
for <span class="emphasis"><em>k</em></span> successes out of <span class="emphasis"><em>N</em></span> trials.
However our confidence in that estimate will be shaped by how many trials
were conducted, and how many successes were observed. The static member
functions <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_upper_bound_on_p</span></code>
allow you to calculate the confidence intervals for your estimate of
the occurrence frequency.
</p>
<p>
The sample program <a href="../../../../../../example/binomial_confidence_limits.cpp" target="_top">binomial_confidence_limits.cpp</a>
illustrates their use. It begins by defining a procedure that will print
a table of confidence limits for various degrees of certainty:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">void</span> <span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">//</span>
<span class="comment">// trials = Total number of trials.</span>
<span class="comment">// successes = Total number of observed successes.</span>
<span class="comment">//</span>
<span class="comment">// Calculate confidence limits for an observed</span>
<span class="comment">// frequency of occurrence that follows a binomial</span>
<span class="comment">// distribution.</span>
<span class="comment">//</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="comment">// Print out general info:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"___________________________________________\n"</span>
<span class="string">"2-Sided Confidence Limits For Success Ratio\n"</span>
<span class="string">"___________________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">trials</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of successes"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample frequency of occurrence"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="keyword">double</span><span class="special">(</span><span class="identifier">successes</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">trials</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
The procedure now defines a table of significance levels: these are the
probabilities that the true occurrence frequency lies outside the calculated
interval:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Some pretty printing of the table header follows:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
<span class="string">"_______________________________________________________________________\n"</span>
<span class="string">"Confidence Lower CP Upper CP Lower JP Upper JP\n"</span>
<span class="string">" Value (%) Limit Limit Limit Limit\n"</span>
<span class="string">"_______________________________________________________________________\n"</span><span class="special">;</span>
</pre>
<p>
And now for the important part - the intervals themselves - for each
value of <span class="emphasis"><em>alpha</em></span>, we call <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">find_lower_upper_on_p</span></code>
to obtain lower and upper bounds respectively. Note that since we are
calculating a two-sided interval, we must divide the value of alpha in
two.
</p>
<p>
Please note that calculating two separate <span class="emphasis"><em>single sided bounds</em></span>,
each with risk level &#945; &#160;is not the same thing as calculating a two sided
interval. Had we calculate two single-sided intervals each with a risk
that the true value is outside the interval of &#945;, then:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
The risk that it is less than the lower bound is &#945;.
</li></ul></div>
<p>
and
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
The risk that it is greater than the upper bound is also &#945;.
</li></ul></div>
<p>
So the risk it is outside <span class="bold"><strong>upper or lower bound</strong></span>,
is <span class="bold"><strong>twice</strong></span> alpha, and the probability
that it is inside the bounds is therefore not nearly as high as one might
have thought. This is why &#945;/2 must be used in the calculations below.
</p>
<p>
In contrast, had we been calculating a single-sided interval, for example:
<span class="emphasis"><em>"Calculate a lower bound so that we are P% sure that the
true occurrence frequency is greater than some value"</em></span>
then we would <span class="bold"><strong>not</strong></span> have divided by two.
</p>
<p>
Finally note that <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>
provides a choice of two methods for the calculation, we print out the
results from both methods in this example:
</p>
<pre class="programlisting"> <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// Calculate Clopper Pearson bounds:</span>
<span class="keyword">double</span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">u</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
<span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">);</span>
<span class="comment">// Print Clopper Pearson Limits:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">u</span><span class="special">;</span>
<span class="comment">// Calculate Jeffreys Prior Bounds:</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">,</span>
<span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span>
<span class="identifier">u</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
<span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">,</span>
<span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span>
<span class="comment">// Print Jeffreys Prior Limits:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">u</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
And that's all there is to it. Let's see some sample output for a 2 in
10 success ratio, first for 20 trials:
</p>
<pre class="programlisting">___________________________________________
2-Sided Confidence Limits For Success Ratio
___________________________________________
Number of Observations = 20
Number of successes = 4
Sample frequency of occurrence = 0.2
_______________________________________________________________________
Confidence Lower CP Upper CP Lower JP Upper JP
Value (%) Limit Limit Limit Limit
_______________________________________________________________________
50.000 0.12840 0.29588 0.14974 0.26916
75.000 0.09775 0.34633 0.11653 0.31861
90.000 0.07135 0.40103 0.08734 0.37274
95.000 0.05733 0.43661 0.07152 0.40823
99.000 0.03576 0.50661 0.04655 0.47859
99.900 0.01905 0.58632 0.02634 0.55960
99.990 0.01042 0.64997 0.01530 0.62495
99.999 0.00577 0.70216 0.00901 0.67897
</pre>
<p>
As you can see, even at the 95% confidence level the bounds are really
quite wide (this example is chosen to be easily compared to the one in
the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a> <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm" target="_top">here</a>).
Note also that the Clopper-Pearson calculation method (CP above) produces
quite noticeably more pessimistic estimates than the Jeffreys Prior method
(JP above).
</p>
<p>
Compare that with the program output for 2000 trials:
</p>
<pre class="programlisting">___________________________________________
2-Sided Confidence Limits For Success Ratio
___________________________________________
Number of Observations = 2000
Number of successes = 400
Sample frequency of occurrence = 0.2000000
_______________________________________________________________________
Confidence Lower CP Upper CP Lower JP Upper JP
Value (%) Limit Limit Limit Limit
_______________________________________________________________________
50.000 0.19382 0.20638 0.19406 0.20613
75.000 0.18965 0.21072 0.18990 0.21047
90.000 0.18537 0.21528 0.18561 0.21503
95.000 0.18267 0.21821 0.18291 0.21796
99.000 0.17745 0.22400 0.17769 0.22374
99.900 0.17150 0.23079 0.17173 0.23053
99.990 0.16658 0.23657 0.16681 0.23631
99.999 0.16233 0.24169 0.16256 0.24143
</pre>
<p>
Now even when the confidence level is very high, the limits are really
quite close to the experimentally calculated value of 0.2. Furthermore
the difference between the two calculation methods is now really quite
small.
</p>
</div>
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<a name="math_toolkit.stat_tut.weg.binom_eg.binom_size_eg"></a><a class="link" href="binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution.">Estimating
Sample Sizes for a Binomial Distribution.</a>
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<p>
Imagine you have a critical component that you know will fail in 1 in
N "uses" (for some suitable definition of "use").
You may want to schedule routine replacement of the component so that
its chance of failure between routine replacements is less than P%. If
the failures follow a binomial distribution (each time the component
is "used" it either fails or does not) then the static member
function <code class="computeroutput"><span class="identifier">binomial_distibution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_maximum_number_of_trials</span></code>
can be used to estimate the maximum number of "uses" of that
component for some acceptable risk level <span class="emphasis"><em>alpha</em></span>.
</p>
<p>
The example program <a href="../../../../../../example/binomial_sample_sizes.cpp" target="_top">binomial_sample_sizes.cpp</a>
demonstrates its usage. It centres on a routine that prints out a table
of maximum sample sizes for various probability thresholds:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_max_sample_size</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success ratio.</span>
<span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span> <span class="comment">// Total number of observed successes permitted.</span>
<span class="special">{</span>
</pre>
<p>
The routine then declares a table of probability thresholds: these are
the maximum acceptable probability that <span class="emphasis"><em>successes</em></span>
or fewer events will be observed. In our example, <span class="emphasis"><em>successes</em></span>
will be always zero, since we want no component failures, but in other
situations non-zero values may well make sense.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Much of the rest of the program is pretty-printing, the important part
is in the calculation of maximum number of permitted trials for each
value of alpha:
</p>
<pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// calculate trials:</span>
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">binomial</span><span class="special">::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
<span class="identifier">successes</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
<span class="comment">// Print Trials:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
Note that since we're calculating the maximum number of trials permitted,
we'll err on the safe side and take the floor of the result. Had we been
calculating the <span class="emphasis"><em>minimum</em></span> number of trials required
to observe a certain number of <span class="emphasis"><em>successes</em></span> using
<code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
we would have taken the ceiling instead.
</p>
<p>
We'll finish off by looking at some sample output, firstly for a 1 in
1000 chance of component failure with each use:
</p>
<pre class="programlisting">________________________
Maximum Number of Trials
________________________
Success ratio = 0.001
Maximum Number of "successes" permitted = 0
____________________________
Confidence Max Number
Value (%) Of Trials
____________________________
50.000 692
75.000 287
90.000 105
95.000 51
99.000 10
99.900 0
99.990 0
99.999 0
</pre>
<p>
So 51 "uses" of the component would yield a 95% chance that
no component failures would be observed.
</p>
<p>
Compare that with a 1 in 1 million chance of component failure:
</p>
<pre class="programlisting">________________________
Maximum Number of Trials
________________________
Success ratio = 0.0000010
Maximum Number of "successes" permitted = 0
____________________________
Confidence Max Number
Value (%) Of Trials
____________________________
50.000 693146
75.000 287681
90.000 105360
95.000 51293
99.000 10050
99.900 1000
99.990 100
99.999 10
</pre>
<p>
In this case, even 1000 uses of the component would still yield a less
than 1 in 1000 chance of observing a component failure (i.e. a 99.9%
chance of no failure).
</p>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_coinflip_example"></a><a class="link" href="binomial_coinflip_example.html" title="Binomial Coin-Flipping Example">Binomial
Coin-Flipping Example</a>
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<p>
An example of a <a href="http://en.wikipedia.org/wiki/Bernoulli_process" target="_top">Bernoulli
process</a> is coin flipping. A variable in such a sequence may be
called a Bernoulli variable.
</p>
<p>
This example shows using the Binomial distribution to predict the probability
of heads and tails when throwing a coin.
</p>
<p>
The number of correct answers (say heads), X, is distributed as a binomial
random variable with binomial distribution parameters number of trials
(flips) n = 10 and probability (success_fraction) of getting a head p
= 0.5 (a 'fair' coin).
</p>
<p>
(Our coin is assumed fair, but we could easily change the success_fraction
parameter p from 0.5 to some other value to simulate an unfair coin,
say 0.6 for one with chewing gum on the tail, so it is more likely to
fall tails down and heads up).
</p>
<p>
First we need some includes and using statements to be able to use the
binomial distribution, some std input and output, and get started:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Using Binomial distribution to predict how many heads and tails."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">try</span>
<span class="special">{</span>
</pre>
<p>
See note <a class="link" href="binomial_coinflip_example.html#coinflip_eg_catch">with the catch block</a>
about why a try and catch block is always a good idea.
</p>
<p>
First, construct a binomial distribution with parameters success_fraction
1/2, and how many flips.
</p>
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// = 50% = 1/2 for a 'fair' coin.</span>
<span class="keyword">int</span> <span class="identifier">flips</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span>
<span class="identifier">binomial</span> <span class="identifier">flip</span><span class="special">(</span><span class="identifier">flips</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
</pre>
<p>
Then some examples of using Binomial moments (and echoing the parameters).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"From "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flips</span> <span class="special">&lt;&lt;</span> <span class="string">" one can expect to get on average "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" heads (or tails)."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Mode is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" correct."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Skewness of binomial distributions is only zero (symmetrical)</span>
<span class="comment">// if success_fraction is exactly one half,</span>
<span class="comment">// for example, when flipping 'fair' coins.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness if success_fraction is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Expect zero for a 'fair' coin.</span>
</pre>
<p>
Now we show a variety of predictions on the probability of heads:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"For "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" coin flips: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting no heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting at least one head is "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">-</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
When we want to calculate the probability for a range or values we can
sum the PDF's:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 0 or 1 heads is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// sum of exactly == probabilities</span>
</pre>
<p>
Or we can use the cdf.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 0 or 1 (&lt;= 1) heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">9</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Note that using
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
is less accurate than using the complement
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Since the subtraction may involve <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">cancellation
error</a>, where as <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">))</span></code>
does not use such a subtraction internally, and so does not exhibit the
problem.
</p>
<p>
To get the probability for a range of heads, we can either add the pdfs
for each number of heads
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span>
<span class="comment">// P(X == 4) + P(X == 5) + P(X == 6)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">5</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
But this is probably less efficient than using the cdf
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span>
<span class="comment">// P(X &lt;= 6) - P(X &lt;= 3) == P(X &lt; 4)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">6</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Certainly for a bigger range like, 3 to 7
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is "</span>
<span class="comment">// P(X &lt;= 7) - P(X &lt;= 2) == P(X &lt; 3)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">7</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Finally, print two tables of probability for the <span class="emphasis"><em>exactly</em></span>
and <span class="emphasis"><em>at least</em></span> a number of heads.
</p>
<pre class="programlisting"><span class="comment">// Print a table of probability for the exactly a number of heads.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly (==) heads"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special">&lt;=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span>
<span class="special">{</span> <span class="comment">// Say success means getting a head (or equally success means getting a tail).</span>
<span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span>
<span class="special">&lt;&lt;</span> <span class="string">", or "</span> <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// for i</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Tabulate the probability of getting between zero heads and 0 upto 10 heads.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting upto (&lt;=) heads"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special">&lt;=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span>
<span class="special">{</span> <span class="comment">// Say success means getting a head</span>
<span class="comment">// (equally success could mean getting a tail).</span>
<span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span> <span class="comment">// P(X &lt;= heads)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span>
<span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">", or "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span><span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// for i</span>
</pre>
<p>
The last (0 to 10 heads) must, of course, be 100% probability.
</p>
<pre class="programlisting"><span class="special">}</span>
<span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">//</span>
</pre>
<p>
<a name="coinflip_eg_catch"></a>It is always essential to include try
&amp; catch blocks because default policies are to throw exceptions on
arguments that are out of domain or cause errors like numeric-overflow.
</p>
<p>
Lacking try &amp; catch blocks, the program will abort, whereas the message
below from the thrown exception will give some helpful clues as to the
cause of the problem.
</p>
<pre class="programlisting"> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"\n"</span><span class="string">"Message from thrown exception was:\n "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
See <a href="../../../../../../example/binomial_coinflip_example.cpp" target="_top">binomial_coinflip_example.cpp</a>
for full source code, the program output looks like this:
</p>
<pre class="programlisting">Using Binomial distribution to predict how many heads and tails.
From 10 one can expect to get on average 5 heads (or tails).
Mode is 5
Standard deviation is 1.581
So about 2/3 will lie within 1 standard deviation and get between 4 and 6 correct.
Skewness is 0
Skewness if success_fraction is 0.5 is 0
For 10 coin flips:
Probability of getting no heads is 0.0009766
Probability of getting at least one head is 0.999
Probability of getting 0 or 1 heads is 0.01074
Probability of getting 0 or 1 (&lt;= 1) heads is 0.01074
Probability of getting 9 or 10 heads is 0.01074
Probability of getting 9 or 10 heads is 0.01074
Probability of getting 9 or 10 heads is 0.01074
Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6562
Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6563
Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is 0.8906
Probability of getting exactly (==) heads
0 0.0009766 or 1 in 1024, or 0.09766%
1 0.009766 or 1 in 102.4, or 0.9766%
2 0.04395 or 1 in 22.76, or 4.395%
3 0.1172 or 1 in 8.533, or 11.72%
4 0.2051 or 1 in 4.876, or 20.51%
5 0.2461 or 1 in 4.063, or 24.61%
6 0.2051 or 1 in 4.876, or 20.51%
7 0.1172 or 1 in 8.533, or 11.72%
8 0.04395 or 1 in 22.76, or 4.395%
9 0.009766 or 1 in 102.4, or 0.9766%
10 0.0009766 or 1 in 1024, or 0.09766%
Probability of getting upto (&lt;=) heads
0 0.0009766 or 1 in 1024, or 0.09766%
1 0.01074 or 1 in 93.09, or 1.074%
2 0.05469 or 1 in 18.29, or 5.469%
3 0.1719 or 1 in 5.818, or 17.19%
4 0.377 or 1 in 2.653, or 37.7%
5 0.623 or 1 in 1.605, or 62.3%
6 0.8281 or 1 in 1.208, or 82.81%
7 0.9453 or 1 in 1.058, or 94.53%
8 0.9893 or 1 in 1.011, or 98.93%
9 0.999 or 1 in 1.001, or 99.9%
10 1 or 1 in 1, or 100%
</pre>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example"></a><a class="link" href="binomial_quiz_example.html" title="Binomial Quiz Example">Binomial
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<p>
A multiple choice test has four possible answers to each of 16 questions.
A student guesses the answer to each question, so the probability of
getting a correct answer on any given question is one in four, a quarter,
1/4, 25% or fraction 0.25. The conditions of the binomial experiment
are assumed to be met: n = 16 questions constitute the trials; each question
results in one of two possible outcomes (correct or incorrect); the probability
of being correct is 0.25 and is constant if no knowledge about the subject
is assumed; the questions are answered independently if the student's
answer to a question in no way influences his/her answer to another question.
</p>
<p>
First, we need to be able to use the binomial distribution constructor
(and some std input/output, of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ios</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">flush</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">exception</span><span class="special">&gt;</span>
</pre>
<p>
The number of correct answers, X, is distributed as a binomial random
variable with binomial distribution parameters: questions n and success
fraction probability p. So we construct a binomial distribution:
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">questions</span> <span class="special">=</span> <span class="number">16</span><span class="special">;</span> <span class="comment">// All the questions in the quiz.</span>
<span class="keyword">int</span> <span class="identifier">answers</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="comment">// Possible answers to each question.</span>
<span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">answers</span><span class="special">;</span> <span class="comment">// If a random guess, p = 1/4 = 0.25.</span>
<span class="identifier">binomial</span> <span class="identifier">quiz</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span>
</pre>
<p>
and display the distribution parameters we used thus:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"In a quiz with "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" questions and with a probability of guessing right of "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">*</span> <span class="number">100</span> <span class="special">&lt;&lt;</span> <span class="string">" %"</span>
<span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;(</span><span class="number">1.</span> <span class="special">/</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">())</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Show a few probabilities of just guessing:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting none right is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.010023</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly one right is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly two right is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">pass_score</span> <span class="special">=</span> <span class="number">11</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by chance is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting all "</span> <span class="special">&lt;&lt;</span> <span class="identifier">questions</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by chance is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">questions</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting none right is 0.0100226
Probability of getting exactly one right is 0.0534538
Probability of getting exactly two right is 0.133635
Probability of getting exactly 11 right is 0.000247132
Probability of getting exactly all 16 answers right by chance is 2.32831e-010
</pre>
<p>
These don't give any encouragement to guessers!
</p>
<p>
We can tabulate the 'getting exactly right' ( == ) probabilities thus:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"Guessed Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special">&lt;=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span>
<span class="special">{</span>
<span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Guessed Probability
0 0.0100226
1 0.0534538
2 0.133635
3 0.207876
4 0.225199
5 0.180159
6 0.110097
7 0.0524273
8 0.0196602
9 0.00582526
10 0.00135923
11 0.000247132
12 3.43239e-005
13 3.5204e-006
14 2.51457e-007
15 1.11759e-008
16 2.32831e-010
</pre>
<p>
Then we can add the probabilities of some 'exactly right' like this:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting none or one right is 0.0634764
</pre>
<p>
But if more than a couple of scores are involved, it is more convenient
(and may be more accurate) to use the Cumulative Distribution Function
(cdf) instead:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting none or one right is 0.0634764
</pre>
<p>
Since the cdf is inclusive, we can get the probability of getting up
to 10 right ( &lt;= )
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting &lt;= 10 right (to fail) is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting &lt;= 10 right (to fail) is 0.999715
</pre>
<p>
To get the probability of getting 11 or more right (to pass), it is tempting
to use
</p>
<pre class="programlisting"><span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span></pre>
<p>
to get the probability of &gt; 10
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting &gt; 10 right (to pass) is "</span> <span class="special">&lt;&lt;</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting &gt; 10 right (to pass) is 0.000285239
</pre>
<p>
But this should be resisted in favor of using the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>
function (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting &gt; 10 right (to pass) is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting &gt; 10 right (to pass) is 0.000285239
</pre>
<p>
And we can check that these two, &lt;= 10 and &gt; 10, add up to unity.
</p>
<pre class="programlisting"><span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1.</span><span class="special">);</span>
</pre>
<p>
If we want a &lt; rather than a &lt;= test, because the CDF is inclusive,
we must subtract one from the score.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting less than "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span>
<span class="special">&lt;&lt;</span> <span class="string">" (&lt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">") answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of getting less than 11 (&lt; 11) answers right by guessing is 0.999715
</pre>
<p>
and similarly to get a &gt;= rather than a &gt; test we also need to
subtract one from the score (and can again check the sum is unity). This
is because if the cdf is <span class="emphasis"><em>inclusive</em></span>, then its complement
must be <span class="emphasis"><em>exclusive</em></span> otherwise there would be one possible
outcome counted twice!
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting at least "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span>
<span class="special">&lt;&lt;</span> <span class="string">"(&gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">") answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span>
<span class="special">&lt;&lt;</span> <span class="string">", only 1 in "</span> <span class="special">&lt;&lt;</span> <span class="number">1</span><span class="special">/</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1</span><span class="special">);</span>
</pre>
<pre class="programlisting">Probability of getting at least 11 (&gt;= 11) answers right by guessing is 0.000285239, only 1 in 3505.83
</pre>
<p>
Finally we can tabulate some probabilities:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"At most (&lt;=)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special">&lt;=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">score</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">At most (&lt;=)
Guessed OK Probability
0 0.01002259576
1 0.0634764398
2 0.1971110499
3 0.4049871101
4 0.6301861752
5 0.8103454274
6 0.9204427481
7 0.9728700437
8 0.9925302796
9 0.9983555346
10 0.9997147608
11 0.9999618928
12 0.9999962167
13 0.9999997371
14 0.9999999886
15 0.9999999998
16 1
</pre>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"At least (&gt;)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special">&lt;=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">score</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<pre class="programlisting">At least (&gt;)
Guessed OK Probability
0 0.9899774042
1 0.9365235602
2 0.8028889501
3 0.5950128899
4 0.3698138248
5 0.1896545726
6 0.07955725188
7 0.02712995629
8 0.00746972044
9 0.001644465374
10 0.0002852391917
11 3.810715862e-005
12 3.783265129e-006
13 2.628657967e-007
14 1.140870154e-008
15 2.328306437e-010
16 0
</pre>
<p>
We now consider the probabilities of <span class="bold"><strong>ranges</strong></span>
of correct guesses.
</p>
<p>
First, calculate the probability of getting a range of guesses right,
by adding the exact probabilities of each from low ... high.
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">low</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span> <span class="comment">// Getting at least 3 right.</span>
<span class="keyword">int</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span> <span class="comment">// Getting as most 5 right.</span>
<span class="keyword">double</span> <span class="identifier">sum</span> <span class="special">=</span> <span class="number">0.</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="identifier">low</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;=</span> <span class="identifier">high</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">sum</span> <span class="special">+=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">i</span><span class="special">);</span>
<span class="special">}</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">sum</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span>
</pre>
<pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132
</pre>
<p>
Or, usually better, we can use the difference of cdfs instead:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span>
</pre>
<pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132
</pre>
<p>
And we can also try a few more combinations of high and low choices:
</p>
<pre class="programlisting"><span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 and 6 P= 0.91042</span>
<span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">8</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 &lt;= x 8 P = 0.9825</span>
<span class="identifier">low</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 4 &lt;= x 4 P = 0.22520</span>
</pre>
<pre class="programlisting">Probability of getting between 1 and 6 answers right by guessing is 0.9104
Probability of getting between 1 and 8 answers right by guessing is 0.9825
Probability of getting between 4 and 4 answers right by guessing is 0.2252
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.using_binomial_distribution_mome"></a></span><a class="link" href="binomial_quiz_example.html#math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.using_binomial_distribution_mome">Using
Binomial distribution moments</a>
</h5>
<p>
Using moments of the distribution, we can say more about the spread of
results from guessing.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"By guessing, on average, one can expect to get "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" correct answers."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" correct."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Mode (the most frequent) is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">By guessing, on average, one can expect to get 4 correct answers.
Standard deviation is 1.732
So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
Mode (the most frequent) is 4
Skewness is 0.2887
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h1"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.quantiles"></a></span><a class="link" href="binomial_quiz_example.html#math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.quantiles">Quantiles</a>
</h5>
<p>
The quantiles (percentiles or percentage points) for a few probability
levels:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quartiles "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"1 standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Deciles "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"5 to 95% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95%</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"2.5 to 97.5% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5%</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"2 to 98% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98%</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If guessing then percentiles 1 to 99% will get "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">" to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" right."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Notice that these output integral values because the default policy is
<code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>.
</p>
<pre class="programlisting">Quartiles 2 to 5
1 standard deviation 2 to 5
Deciles 1 to 6
5 to 95% 0 to 7
2.5 to 97.5% 0 to 8
2 to 98% 0 to 8
</pre>
<p>
Quantiles values are controlled by the <a class="link" href="../../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding
discrete quantiles</a> quantile policy chosen. The default is <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>, so the lower
quantile is rounded down, and the upper quantile is rounded up.
</p>
<p>
But we might believe that the real values tell us a little more - see
<a class="link" href="../../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">discrete functions</a>.
</p>
<p>
We could control the policy for <span class="bold"><strong>all</strong></span>
distributions by
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span>
<span class="identifier">at</span> <span class="identifier">the</span> <span class="identifier">head</span> <span class="identifier">of</span> <span class="identifier">the</span> <span class="identifier">program</span> <span class="identifier">would</span> <span class="identifier">make</span> <span class="keyword">this</span> <span class="identifier">policy</span> <span class="identifier">apply</span>
</pre>
<p>
to this <span class="bold"><strong>one, and only</strong></span>, translation unit.
</p>
<p>
Or we can now create a (typedef for) policy that has discrete quantiles
real (here avoiding any 'using namespaces ...' statements):
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">real</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">integer_round_outwards</span><span class="special">;</span> <span class="comment">// Default.</span>
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">real</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">real_quantile_policy</span><span class="special">;</span>
</pre>
<p>
Add a custom binomial distribution called
</p>
<pre class="programlisting"><span class="identifier">real_quantile_binomial</span></pre>
<p>
that uses
</p>
<pre class="programlisting"><span class="identifier">real_quantile_policy</span></pre>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">real_quantile_policy</span><span class="special">&gt;</span> <span class="identifier">real_quantile_binomial</span><span class="special">;</span>
</pre>
<p>
Construct an object of this custom distribution:
</p>
<pre class="programlisting"><span class="identifier">real_quantile_binomial</span> <span class="identifier">quiz_real</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span>
</pre>
<p>
And use this to show some quantiles - that now have real rather than
integer values.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quartiles "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles 2 to 4.6212</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"1 standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd 2.6654 4.194</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Deciles "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles 1.3487 5.7583</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"5 to 95% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95% 0.83739 6.4559</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"2.5 to 97.5% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5% 0.42806 7.0688</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"2 to 98% "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98% 0.31311 7.7880</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If guessing, then percentiles 1 to 99% will get "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">" to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" right."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Real Quantiles
Quartiles 2 to 4.621
1 standard deviation 2.665 to 4.194
Deciles 1.349 to 5.758
5 to 95% 0.8374 to 6.456
2.5 to 97.5% 0.4281 to 7.069
2 to 98% 0.3131 to 7.252
If guessing then percentiles 1 to 99% will get 0 to 7.788 right.
</pre>
<p>
See <a href="../../../../../../example/binomial_quiz_example.cpp" target="_top">binomial_quiz_example.cpp</a>
for full source code and output.
</p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<dt><span class="section"><a href="cs_eg/chi_sq_intervals.html">Confidence
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<a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals"></a><a class="link" href="chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation">Confidence
Intervals on the Standard Deviation</a>
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<p>
Once you have calculated the standard deviation for your data, a legitimate
question to ask is "How reliable is the calculated standard deviation?".
For this situation the Chi Squared distribution can be used to calculate
confidence intervals for the standard deviation.
</p>
<p>
The full example code &amp; sample output is in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>.
</p>
<p>
We'll begin by defining the procedure that will calculate and print out
the confidence intervals:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">confidence_limits_on_std_deviation</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="comment">// Sample Standard Deviation</span>
<span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">)</span> <span class="comment">// Sample size</span>
<span class="special">{</span>
</pre>
<p>
We'll begin by printing out some general information:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"________________________________________________\n"</span>
<span class="string">"2-Sided Confidence Limits For Standard Deviation\n"</span>
<span class="string">"________________________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
and then define a table of significance levels for which we'll calculate
intervals:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
The distribution we'll need to calculate the confidence intervals is
a Chi Squared distribution, with N-1 degrees of freedom:
</p>
<pre class="programlisting"><span class="identifier">chi_squared</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
For each value of alpha, the formula for the confidence interval is given
by:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut1.svg"></span>
</p>
<p>
Where <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut2.svg"></span> is the upper critical value, and <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut3.svg"></span> is
the lower critical value of the Chi Squared distribution.
</p>
<p>
In code we begin by printing out a table header:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
<span class="string">"_____________________________________________\n"</span>
<span class="string">"Confidence Lower Upper\n"</span>
<span class="string">" Value (%) Limit Limit\n"</span>
<span class="string">"_____________________________________________\n"</span><span class="special">;</span>
</pre>
<p>
and then loop over the values of alpha and calculate the intervals for
each: remember that the lower critical value is the same as the quantile,
and the upper critical value is the same as the quantile from the complement
of the probability:
</p>
<pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// Calculate limits:</span>
<span class="keyword">double</span> <span class="identifier">lower_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">)));</span>
<span class="keyword">double</span> <span class="identifier">upper_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
<span class="comment">// Print Limits:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">lower_limit</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">upper_limit</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
To see some example output we'll use the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm" target="_top">gear
data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>. The data represents measurements
of gear diameter from a manufacturing process.
</p>
<pre class="programlisting">________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Number of Observations = 100
Standard Deviation = 0.006278908
_____________________________________________
Confidence Lower Upper
Value (%) Limit Limit
_____________________________________________
50.000 0.00601 0.00662
75.000 0.00582 0.00685
90.000 0.00563 0.00712
95.000 0.00551 0.00729
99.000 0.00530 0.00766
99.900 0.00507 0.00812
99.990 0.00489 0.00855
99.999 0.00474 0.00895
</pre>
<p>
So at the 95% confidence level we conclude that the standard deviation
is between 0.00551 and 0.00729.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_functi"></a></span><a class="link" href="chi_sq_intervals.html#math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_functi">Confidence
intervals as a function of the number of observations</a>
</h5>
<p>
Similarly, we can also list the confidence intervals for the standard
deviation for the common confidence levels 95%, for increasing numbers
of observations.
</p>
<p>
The standard deviation used to compute these values is unity, so the
limits listed are <span class="bold"><strong>multipliers</strong></span> for any
particular standard deviation. For example, given a standard deviation
of 0.0062789 as in the example above; for 100 observations the multiplier
is 0.8780 giving the lower confidence limit of 0.8780 * 0.006728 = 0.00551.
</p>
<pre class="programlisting">____________________________________________________
Confidence level (two-sided) = 0.0500000
Standard Deviation = 1.0000000
________________________________________
Observations Lower Upper
Limit Limit
________________________________________
2 0.4461 31.9102
3 0.5207 6.2847
4 0.5665 3.7285
5 0.5991 2.8736
6 0.6242 2.4526
7 0.6444 2.2021
8 0.6612 2.0353
9 0.6755 1.9158
10 0.6878 1.8256
15 0.7321 1.5771
20 0.7605 1.4606
30 0.7964 1.3443
40 0.8192 1.2840
50 0.8353 1.2461
60 0.8476 1.2197
100 0.8780 1.1617
120 0.8875 1.1454
1000 0.9580 1.0459
10000 0.9863 1.0141
50000 0.9938 1.0062
100000 0.9956 1.0044
1000000 0.9986 1.0014
</pre>
<p>
With just 2 observations the limits are from <span class="bold"><strong>0.445</strong></span>
up to to <span class="bold"><strong>31.9</strong></span>, so the standard deviation
might be about <span class="bold"><strong>half</strong></span> the observed value
up to <span class="bold"><strong>30 times</strong></span> the observed value!
</p>
<p>
Estimating a standard deviation with just a handful of values leaves
a very great uncertainty, especially the upper limit. Note especially
how far the upper limit is skewed from the most likely standard deviation.
</p>
<p>
Even for 10 observations, normally considered a reasonable number, the
range is still from 0.69 to 1.8, about a range of 0.7 to 2, and is still
highly skewed with an upper limit <span class="bold"><strong>twice</strong></span>
the median.
</p>
<p>
When we have 1000 observations, the estimate of the standard deviation
is starting to look convincing, with a range from 0.95 to 1.05 - now
near symmetrical, but still about + or - 5%.
</p>
<p>
Only when we have 10000 or more repeated observations can we start to
be reasonably confident (provided we are sure that other factors like
drift are not creeping in).
</p>
<p>
For 10000 observations, the interval is 0.99 to 1.1 - finally a really
convincing + or -1% confidence.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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<a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_size"></a><a class="link" href="chi_sq_size.html" title="Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation">Estimating
the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</a>
</h5></div></div></div>
<p>
Suppose we conduct a Chi Squared test for standard deviation and the
result is borderline, a legitimate question to ask is "How large
would the sample size have to be in order to produce a definitive result?"
</p>
<p>
The class template <a class="link" href="../../../dist_ref/dists/chi_squared_dist.html" title="Chi Squared Distribution">chi_squared_distribution</a>
has a static method <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
that will calculate this value for some acceptable risk of type I failure
<span class="emphasis"><em>alpha</em></span>, type II failure <span class="emphasis"><em>beta</em></span>,
and difference from the standard deviation <span class="emphasis"><em>diff</em></span>.
Please note that the method used works on variance, and not standard
deviation as is usual for the Chi Squared Test.
</p>
<p>
The code for this example is located in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>.
</p>
<p>
We begin by defining a procedure to print out the sample sizes required
for various risk levels:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_sample_sized</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">diff</span><span class="special">,</span> <span class="comment">// difference from variance to detect</span>
<span class="keyword">double</span> <span class="identifier">variance</span><span class="special">)</span> <span class="comment">// true variance</span>
<span class="special">{</span>
</pre>
<p>
The procedure begins by printing out the input data:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="comment">// Print out general info:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"_____________________________________________________________\n"</span>
<span class="string">"Estimated sample sizes required for various confidence levels\n"</span>
<span class="string">"_____________________________________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"True Variance"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">variance</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Difference to detect"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">diff</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
And defines a table of significance levels for which we'll calculate
sample sizes:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
For each value of alpha we can calculate two sample sizes: one where
the sample variance is less than the true value by <span class="emphasis"><em>diff</em></span>
and one where it is greater than the true value by <span class="emphasis"><em>diff</em></span>.
Thanks to the asymmetric nature of the Chi Squared distribution these
two values will not be the same, the difference in their calculation
differs only in the sign of <span class="emphasis"><em>diff</em></span> that's passed to
<code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>.
Finally in this example we'll simply things, and let risk level <span class="emphasis"><em>beta</em></span>
be the same as <span class="emphasis"><em>alpha</em></span>:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
<span class="string">"_______________________________________________________________\n"</span>
<span class="string">"Confidence Estimated Estimated\n"</span>
<span class="string">" Value (%) Sample Size Sample Size\n"</span>
<span class="string">" (lower one (upper one\n"</span>
<span class="string">" sided test) sided test)\n"</span>
<span class="string">"_______________________________________________________________\n"</span><span class="special">;</span>
<span class="comment">//</span>
<span class="comment">// Now print out the data for the table rows.</span>
<span class="comment">//</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// calculate df for a lower single sided test:</span>
<span class="keyword">double</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">chi_squared</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
<span class="special">-</span><span class="identifier">diff</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">variance</span><span class="special">);</span>
<span class="comment">// convert to sample size:</span>
<span class="keyword">double</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
<span class="comment">// Print size:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span><span class="special">;</span>
<span class="comment">// calculate df for an upper single sided test:</span>
<span class="identifier">df</span> <span class="special">=</span> <span class="identifier">chi_squared</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
<span class="identifier">diff</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">variance</span><span class="special">);</span>
<span class="comment">// convert to sample size:</span>
<span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
<span class="comment">// Print size:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
For some example output, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon
wafer data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>. In this scenario a supplier
of 100 ohm.cm silicon wafers claims that his fabrication process can
produce wafers with sufficient consistency so that the standard deviation
of resistivity for the lot does not exceed 10 ohm.cm. A sample of N =
10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm,
and the question we ask ourselves is "How large would our sample
have to be to reliably detect this difference?".
</p>
<p>
To use our procedure above, we have to convert the standard deviations
to variance (square them), after which the program output looks like
this:
</p>
<pre class="programlisting">_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Variance = 100.00000
Difference to detect = 95.16090
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(lower one (upper one
sided test) sided test)
_______________________________________________________________
50.000 2 2
75.000 2 10
90.000 4 32
95.000 5 51
99.000 7 99
99.900 11 174
99.990 15 251
99.999 20 330
</pre>
<p>
In this case we are interested in a upper single sided test. So for example,
if the maximum acceptable risk of falsely rejecting the null-hypothesis
is 0.05 (Type I error), and the maximum acceptable risk of failing to
reject the null-hypothesis is also 0.05 (Type II error), we estimate
that we would need a sample size of 51.
</p>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_test"></a><a class="link" href="chi_sq_test.html" title="Chi-Square Test for the Standard Deviation">Chi-Square
Test for the Standard Deviation</a>
</h5></div></div></div>
<p>
We use this test to determine whether the standard deviation of a sample
differs from a specified value. Typically this occurs in process change
situations where we wish to compare the standard deviation of a new process
to an established one.
</p>
<p>
The code for this example is contained in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>,
and we'll begin by defining the procedure that will print out the test
statistics:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_test</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="comment">// Sample std deviation</span>
<span class="keyword">double</span> <span class="identifier">D</span><span class="special">,</span> <span class="comment">// True std deviation</span>
<span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">,</span> <span class="comment">// Sample size</span>
<span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// Significance level</span>
<span class="special">{</span>
</pre>
<p>
The procedure begins by printing a summary of the input data:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="comment">// Print header:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"______________________________________________\n"</span>
<span class="string">"Chi Squared test for sample standard deviation\n"</span>
<span class="string">"______________________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Expected True Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
The test statistic (T) is simply the ratio of the sample and "true"
standard deviations squared, multiplied by the number of degrees of freedom
(the sample size less one):
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">D</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">D</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Test Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
The distribution we need to use, is a Chi Squared distribution with N-1
degrees of freedom:
</p>
<pre class="programlisting"><span class="identifier">chi_squared</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
The various hypothesis that can be tested are summarised in the following
table:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Hypothesis
</p>
</th>
<th>
<p>
Test
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
The null-hypothesis: there is no difference in standard deviation
from the specified value
</p>
</td>
<td>
<p>
Reject if T &lt; &#967;<sup>2</sup><sub>(1-alpha/2; N-1)</sub> or T &gt; &#967;<sup>2</sup><sub>(alpha/2; N-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: there is a difference in standard
deviation from the specified value
</p>
</td>
<td>
<p>
Reject if &#967;<sup>2</sup><sub>(1-alpha/2; N-1)</sub> &gt;= T &gt;= &#967;<sup>2</sup><sub>(alpha/2; N-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: the standard deviation is less
than the specified value
</p>
</td>
<td>
<p>
Reject if &#967;<sup>2</sup><sub>(1-alpha; N-1)</sub> &lt;= T
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: the standard deviation is greater
than the specified value
</p>
</td>
<td>
<p>
Reject if &#967;<sup>2</sup><sub>(alpha; N-1)</sub> &gt;= T
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
Where &#967;<sup>2</sup><sub>(alpha; N-1)</sub> is the upper critical value of the Chi Squared distribution,
and &#967;<sup>2</sup><sub>(1-alpha; N-1)</sub> is the lower critical value.
</p>
<p>
Recall that the lower critical value is the same as the quantile, and
the upper critical value is the same as the quantile from the complement
of the probability, that gives us the following code to calculate the
critical values:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ucv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">));</span>
<span class="keyword">double</span> <span class="identifier">ucv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
<span class="keyword">double</span> <span class="identifier">lcv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">lcv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
Now that we have the critical values, we can compare these to our test
statistic, and print out the result of each hypothesis and test:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span>
<span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation != "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special">&lt;</span> <span class="identifier">t_stat</span><span class="special">)</span> <span class="special">||</span> <span class="special">(</span><span class="identifier">lcv2</span> <span class="special">&gt;</span> <span class="identifier">t_stat</span><span class="special">))</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation &lt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">&gt;</span> <span class="identifier">t_stat</span><span class="special">)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special">&lt;</span> <span class="identifier">t_stat</span><span class="special">)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
To see some example output we'll use the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm" target="_top">gear
data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>. The data represents measurements
of gear diameter from a manufacturing process. The program output is
deliberately designed to mirror the DATAPLOT output shown in the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm" target="_top">NIST
Handbook Example</a>.
</p>
<pre class="programlisting">______________________________________________
Chi Squared test for sample standard deviation
______________________________________________
Number of Observations = 100
Sample Standard Deviation = 0.00628
Expected True Standard Deviation = 0.10000
Test Statistic = 0.39030
CDF of test statistic: = 1.438e-099
Upper Critical Value at alpha: = 1.232e+002
Upper Critical Value at alpha/2: = 1.284e+002
Lower Critical Value at alpha: = 7.705e+001
Lower Critical Value at alpha/2: = 7.336e+001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard Deviation != 0.100 ACCEPTED
Standard Deviation &lt; 0.100 ACCEPTED
Standard Deviation &gt; 0.100 REJECTED
</pre>
<p>
In this case we are testing whether the sample standard deviation is
0.1, and the null-hypothesis is rejected, so we conclude that the standard
deviation <span class="emphasis"><em>is not</em></span> 0.1.
</p>
<p>
For an alternative example, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon
wafer data</a> again from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>. In this scenario a supplier
of 100 ohm.cm silicon wafers claims that his fabrication process can
produce wafers with sufficient consistency so that the standard deviation
of resistivity for the lot does not exceed 10 ohm.cm. A sample of N =
10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm,
and the question we ask ourselves is "Is the suppliers claim correct?".
</p>
<p>
The program output now looks like this:
</p>
<pre class="programlisting">______________________________________________
Chi Squared test for sample standard deviation
______________________________________________
Number of Observations = 10
Sample Standard Deviation = 13.97000
Expected True Standard Deviation = 10.00000
Test Statistic = 17.56448
CDF of test statistic: = 9.594e-001
Upper Critical Value at alpha: = 1.692e+001
Upper Critical Value at alpha/2: = 1.902e+001
Lower Critical Value at alpha: = 3.325e+000
Lower Critical Value at alpha/2: = 2.700e+000
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard Deviation != 10.000 REJECTED
Standard Deviation &lt; 10.000 REJECTED
Standard Deviation &gt; 10.000 ACCEPTED
</pre>
<p>
In this case, our null-hypothesis is that the standard deviation of the
sample is less than 10: this hypothesis is rejected in the analysis above,
and so we reject the manufacturers claim.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.dist_construct_eg"></a><a class="link" href="dist_construct_eg.html" title="Distribution Construction Examples">Distribution
Construction Examples</a>
</h4></div></div></div>
<p>
The structure of distributions is rather different from some other statistical
libraries, for example, those written in less object-oriented language
like FORTRAN and C: these provide a few arguments to each free function.
</p>
<p>
Boost.Math library provides each distribution as a template C++ class.
A distribution is constructed with a few arguments, and then member and
non-member functions are used to find values of the distribution, often
a function of a random variate.
</p>
<p>
For this demonstration, first we need some includes to access the negative
binomial distribution (and the binomial, beta and gamma distributions too).
</p>
<p>
To demonstrate the use with a high precision User-defined floating-point
type <code class="computeroutput"><span class="identifier">cpp_dec_float</span></code> we also
need an include from Boost.Multiprecision.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for negative_binomial_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> <span class="comment">// default type is double.</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for binomial_distribution.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for beta_distribution.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for gamma_distribution.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for cpp_dec_float_100</span>
</pre>
<p>
Several examples of constructing distributions follow:
</p>
<p>
First, a negative binomial distribution with 8 successes and a success
fraction 0.25, 25% or 1 in 4, is constructed like this:
</p>
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">mydist0</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
</pre>
<p>
But this is inconveniently long, so we might be tempted to write
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
</pre>
<p>
but this might risk ambiguity with names in <code class="computeroutput"><span class="identifier">std</span>
<span class="identifier">random</span></code> so <span class="bold"><strong>much</strong></span>
better is explicit <code class="computeroutput"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span></code>
statements, for example:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span>
</pre>
<p>
and we can still reduce typing.
</p>
<p>
Since the vast majority of applications use will be using <code class="computeroutput"><span class="keyword">double</span></code> precision, the template argument
to the distribution (<code class="computeroutput"><span class="identifier">RealType</span></code>)
defaults to type <code class="computeroutput"><span class="keyword">double</span></code>, so
we can also write:
</p>
<pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">mydist9</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span> <span class="comment">// Uses default `RealType = double`.</span>
</pre>
<p>
But the name <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span></code>
is still inconveniently long, so, for most distributions, a convenience
<code class="computeroutput"><span class="keyword">typedef</span></code> is provided, for example:
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">negative_binomial</span><span class="special">;</span> <span class="comment">// Reserved name of type double.</span>
</pre>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
This convenience typedef is <span class="bold"><strong>not provided</strong></span>
if a clash would occur with the name of a function: currently only <code class="computeroutput"><span class="identifier">beta</span></code> and <code class="computeroutput"><span class="identifier">gamma</span></code>
fall into this category.
</p></td></tr>
</table></div>
<p>
So, after a using statement,
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span>
</pre>
<p>
we have a convenient typedef to <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span></code>:
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">mydist</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
</pre>
<p>
Some more examples using the convenience typedef:
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">mydist10</span><span class="special">(</span><span class="number">5.</span><span class="special">,</span> <span class="number">0.4</span><span class="special">);</span> <span class="comment">// Both arguments double.</span>
</pre>
<p>
And automatic conversion takes place, so you can use integers and floats:
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">mydist11</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">0.4</span><span class="special">);</span> <span class="comment">// Using provided typedef double, int and double arguments.</span>
</pre>
<p>
This is probably the most common usage.
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">mydist12</span><span class="special">(</span><span class="number">5.</span><span class="special">,</span> <span class="number">0.4F</span><span class="special">);</span> <span class="comment">// Double and float arguments.</span>
<span class="identifier">negative_binomial</span> <span class="identifier">mydist13</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Both arguments integer.</span>
</pre>
<p>
Similarly for most other distributions like the binomial.
</p>
<pre class="programlisting"><span class="identifier">binomial</span> <span class="identifier">mybinomial</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// is more concise than</span>
<span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">mybinomd1</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
</pre>
<p>
For cases when the typdef distribution name would clash with a math special
function (currently only beta and gamma) the typedef is deliberately not
provided, and the longer version of the name must be used. For example
do not use:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">beta</span><span class="special">;</span>
<span class="identifier">beta</span> <span class="identifier">mybetad0</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// Error beta is a math FUNCTION!</span>
</pre>
<p>
Which produces the error messages:
</p>
<pre class="programlisting">error C2146: syntax error : missing ';' before identifier 'mybetad0'
warning C4551: function call missing argument list
error C3861: 'mybetad0': identifier not found
</pre>
<p>
Instead you should use:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">beta_distribution</span><span class="special">;</span>
<span class="identifier">beta_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">mybetad1</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
</pre>
<p>
or for the gamma distribution:
</p>
<pre class="programlisting"><span class="identifier">gamma_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">mygammad1</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
</pre>
<p>
We can, of course, still provide the type explicitly thus:
</p>
<pre class="programlisting"><span class="comment">// Explicit double precision: both arguments are double:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">mydist1</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
<span class="comment">// Explicit float precision, double arguments are truncated to float:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">mydist2</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
<span class="comment">// Explicit float precision, integer &amp; double arguments converted to float:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">mydist3</span><span class="special">(</span><span class="number">8</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
<span class="comment">// Explicit float precision, float arguments, so no conversion:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">mydist4</span><span class="special">(</span><span class="number">8.F</span><span class="special">,</span> <span class="number">0.25F</span><span class="special">);</span>
<span class="comment">// Explicit float precision, integer arguments promoted to float.</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">mydist5</span><span class="special">(</span><span class="number">8</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="comment">// Explicit double precision:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">mydist6</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
<span class="comment">// Explicit long double precision:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">mydist7</span><span class="special">(</span><span class="number">8.</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
</pre>
<p>
And you can use your own RealType, for example, <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span></code>
(an arbitrary 50 decimal digits precision type), then we can write:
</p>
<pre class="programlisting"> <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;</span> <span class="identifier">mydist8</span><span class="special">(</span><span class="number">8</span><span class="special">,</span> <span class="number">0.25</span><span class="special">);</span>
<span class="comment">// `integer` arguments are promoted to your RealType exactly, but</span>
<span class="comment">// `double` argument are converted to RealType,</span>
<span class="comment">// possibly losing precision, so don't write:</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;</span> <span class="identifier">mydist20</span><span class="special">(</span><span class="number">8</span><span class="special">,</span> <span class="number">0.23456789012345678901234567890</span><span class="special">);</span>
<span class="comment">// to avoid truncation of second parameter to `0.2345678901234567`.</span>
<span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;</span> <span class="identifier">mydist21</span><span class="special">(</span><span class="number">8</span><span class="special">,</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="string">"0.23456789012345678901234567890"</span><span class="special">)</span> <span class="special">);</span>
<span class="comment">// Ensure that all potentially significant digits are shown.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
<span class="identifier">cpp_dec_float_50</span> <span class="identifier">x</span><span class="special">(</span><span class="string">"1.23456789012345678901234567890"</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">mydist8</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting"><span class="identifier">showing</span> <span class="number">0.00012630010495970320103876754721976419438231705359935</span>
</pre>
<div class="warning"><table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td>
<th align="left">Warning</th>
</tr>
<tr><td align="left" valign="top"><p>
When using multiprecision, it is all too easy to get accidental truncation!
</p></td></tr>
</table></div>
<p>
For example, if you write
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">mydist8</span><span class="special">,</span> <span class="number">1.23456789012345678901234567890</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
showing 0.00012630010495970318465064569310967179576805651692929, which
is wrong at about the 17th decimal digit!
</p>
<p>
This is because the value provided is truncated to a <code class="computeroutput"><span class="keyword">double</span></code>,
effectively <code class="computeroutput"><span class="keyword">double</span> <span class="identifier">x</span>
<span class="special">=</span> <span class="number">1.23456789012345678901234567890</span><span class="special">;</span></code>
</p>
<p>
Then the now <code class="computeroutput"><span class="keyword">double</span> <span class="identifier">x</span></code>
is passed to function <code class="computeroutput"><span class="identifier">pdf</span></code>,
and this truncated <code class="computeroutput"><span class="keyword">double</span></code>
value is finally promoted to <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>.
</p>
<p>
Another way of quietly getting the wrong answer is to write:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">mydist8</span><span class="special">,</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">1.23456789012345678901234567890</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
A correct way from a multi-digit string value is
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">mydist8</span><span class="special">,</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="string">"1.23456789012345678901234567890"</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
Getting about 17 decimal digits followed by many zeros is often a sign
of accidental truncation.
</p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.stat_tut.weg.dist_construct_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.dist_construct_eg.default_arguments_to_distributio"></a></span><a class="link" href="dist_construct_eg.html#math_toolkit.stat_tut.weg.dist_construct_eg.default_arguments_to_distributio">Default
arguments to distribution constructors.</a>
</h5>
<p>
Note that default constructor arguments are only provided for some distributions.
So if you wrongly assume a default argument, you will get an error message,
for example:
</p>
<pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">mydist8</span><span class="special">;</span>
</pre>
<pre class="programlisting">error C2512 no appropriate default constructor available.</pre>
<p>
No default constructors are provided for the <code class="computeroutput"><span class="identifier">negative</span>
<span class="identifier">binomial</span></code> distribution, because
it is difficult to chose any sensible default values for this distribution.
</p>
<p>
For other distributions, like the normal distribution, it is obviously
very useful to provide 'standard' defaults for the mean (zero) and standard
deviation (unity) thus:
</p>
<pre class="programlisting"><span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
So in this case we can write:
</p>
<pre class="programlisting"> <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">norm1</span><span class="special">;</span> <span class="comment">// Standard normal distribution.</span>
<span class="identifier">normal</span> <span class="identifier">norm2</span><span class="special">(</span><span class="number">2</span><span class="special">);</span> <span class="comment">// Mean = 2, std deviation = 1.</span>
<span class="identifier">normal</span> <span class="identifier">norm3</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span> <span class="comment">// Mean = 2, std deviation = 3.</span>
<span class="special">}</span>
<span class="keyword">catch</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span> <span class="special">&amp;</span><span class="identifier">ex</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// int main()</span>
</pre>
<p>
There is no useful output from this demonstration program, of course.
</p>
<p>
See <a href="../../../../../example/distribution_construction.cpp" target="_top">distribution_construction.cpp</a>
for full source code.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.error_eg"></a><a class="link" href="error_eg.html" title="Error Handling Example">Error Handling
Example</a>
</h4></div></div></div>
<p>
See <a class="link" href="../../error_handling.html" title="Error Handling">error handling documentation</a>
for a detailed explanation of the mechanism of handling errors, including
the common "bad" arguments to distributions and functions, and
how to use <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a> to control it.
</p>
<p>
But, by default, <span class="bold"><strong>exceptions will be raised</strong></span>,
for domain errors, pole errors, numeric overflow, and internal evaluation
errors. To avoid the exceptions from getting thrown and instead get an
appropriate value returned, usually a NaN (domain errors pole errors or
internal errors), or infinity (from overflow), you need to change the policy.
</p>
<p>
The following example demonstrates the effect of setting the macro BOOST_MATH_DOMAIN_ERROR_POLICY
when an invalid argument is encountered. For the purposes of this example,
we'll pass a negative degrees of freedom parameter to the student's t distribution.
</p>
<p>
Since we know that this is a single file program we could just add:
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> <span class="identifier">ignore_error</span>
</pre>
<p>
to the top of the source file to change the default policy to one that
simply returns a NaN when a domain error occurs. Alternatively we could
use:
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> <span class="identifier">errno_on_error</span>
</pre>
<p>
To ensure the <code class="computeroutput"><span class="special">::</span><span class="identifier">errno</span></code>
is set when a domain error occurs as well as returning a NaN.
</p>
<p>
This is safe provided the program consists of a single translation unit
<span class="emphasis"><em>and</em></span> we place the define <span class="emphasis"><em>before</em></span>
any #includes. Note that should we add the define after the includes then
it will have no effect! A warning such as:
</p>
<pre class="programlisting">warning C4005: 'BOOST_MATH_OVERFLOW_ERROR_POLICY' : macro redefinition</pre>
<p>
is a certain sign that it will <span class="emphasis"><em>not</em></span> have the desired
effect.
</p>
<p>
We'll begin our sample program with the needed includes:
</p>
<pre class="programlisting"> <span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> <span class="identifier">ignore_error</span>
<span class="comment">// Boost</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">students_t</span><span class="special">;</span> <span class="comment">// Probability of students_t(df, t).</span>
<span class="comment">// std</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">stdexcept</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">cstddef</span><span class="special">&gt;</span>
<span class="comment">// using ::errno</span>
</pre>
<p>
Next we'll define the program's main() to call the student's t distribution
with an invalid degrees of freedom parameter, the program is set up to
handle either an exception or a NaN:
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Example error handling using Student's t function. "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"BOOST_MATH_DOMAIN_ERROR_POLICY is set to: "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">BOOST_STRINGIZE</span><span class="special">(</span><span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">degrees_of_freedom</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span> <span class="comment">// A bad argument!</span>
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span>
<span class="keyword">try</span>
<span class="special">{</span>
<span class="identifier">errno</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// Clear/reset.</span>
<span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">degrees_of_freedom</span><span class="special">);</span> <span class="comment">// exception is thrown here if enabled.</span>
<span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">t</span><span class="special">);</span>
<span class="comment">// Test for error reported by other means:</span>
<span class="keyword">if</span><span class="special">((</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">isnan</span><span class="special">)(</span><span class="identifier">p</span><span class="special">))</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cdf returned a NaN!"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">errno</span> <span class="special">!=</span> <span class="number">0</span><span class="special">)</span>
<span class="special">{</span> <span class="comment">// So errno has been set.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"errno is set to: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">errno</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="special">}</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of Student's t is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"\n"</span><span class="string">"Message from thrown exception was:\n "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// int main()</span>
</pre>
<p>
Here's what the program output looks like with a default build (one that
<span class="bold"><strong>does throw exceptions</strong></span>):
</p>
<pre class="programlisting">Example error handling using Student's t function.
BOOST_MATH_DOMAIN_ERROR_POLICY is set to: throw_on_error
Message from thrown exception was:
Error in function boost::math::students_t_distribution&lt;double&gt;::students_t_distribution:
Degrees of freedom argument is -1, but must be &gt; 0 !
</pre>
<p>
Alternatively let's build with:
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> <span class="identifier">ignore_error</span>
</pre>
<p>
Now the program output is:
</p>
<pre class="programlisting">Example error handling using Student's t function.
BOOST_MATH_DOMAIN_ERROR_POLICY is set to: ignore_error
cdf returned a NaN!
</pre>
<p>
And finally let's build with:
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> <span class="identifier">errno_on_error</span>
</pre>
<p>
Which gives the output show errno:
</p>
<pre class="programlisting">Example error handling using Student's t function.
BOOST_MATH_DOMAIN_ERROR_POLICY is set to: errno_on_error
cdf returned a NaN!
errno is set to: 33
</pre>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top">
<p>
If throwing of exceptions is enabled (the default) but you do <span class="bold"><strong>not</strong></span> have try &amp; catch block, then the program
will terminate with an uncaught exception and probably abort.
</p>
<p>
Therefore to get the benefit of helpful error messages, enabling <span class="bold"><strong>all exceptions and using try &amp; catch</strong></span> is recommended
for most applications.
</p>
<p>
However, for simplicity, the is not done for most examples.
</p>
</td></tr>
</table></div>
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<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.f_eg"></a><a class="link" href="f_eg.html" title="F Distribution Examples">F Distribution Examples</a>
</h4></div></div></div>
<p>
Imagine that you want to compare the standard deviations of two sample
to determine if they differ in any significant way, in this situation you
use the F distribution and perform an F-test. This situation commonly occurs
when conducting a process change comparison: "is a new process more
consistent that the old one?".
</p>
<p>
In this example we'll be using the data for ceramic strength from <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm" target="_top">http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm</a>.
The data for this case study were collected by Said Jahanmir of the NIST
Ceramics Division in 1996 in connection with a NIST/industry ceramics consortium
for strength optimization of ceramic strength.
</p>
<p>
The example program is <a href="../../../../../example/f_test.cpp" target="_top">f_test.cpp</a>,
program output has been deliberately made as similar as possible to the
DATAPLOT output in the corresponding <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm" target="_top">NIST
EngineeringStatistics Handbook example</a>.
</p>
<p>
We'll begin by defining the procedure to conduct the test:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">f_test</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">sd1</span><span class="special">,</span> <span class="comment">// Sample 1 std deviation</span>
<span class="keyword">double</span> <span class="identifier">sd2</span><span class="special">,</span> <span class="comment">// Sample 2 std deviation</span>
<span class="keyword">double</span> <span class="identifier">N1</span><span class="special">,</span> <span class="comment">// Sample 1 size</span>
<span class="keyword">double</span> <span class="identifier">N2</span><span class="special">,</span> <span class="comment">// Sample 2 size</span>
<span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// Significance level</span>
<span class="special">{</span>
</pre>
<p>
The procedure begins by printing out a summary of our input data:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="comment">// Print header:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"____________________________________\n"</span>
<span class="string">"F test for equal standard deviations\n"</span>
<span class="string">"____________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample 1:\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N1</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd1</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample 2:\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
The test statistic for an F-test is simply the ratio of the square of the
two standard deviations:
</p>
<p>
F = s<sub>1</sub><sup>2</sup> / s<sub>2</sub><sup>2</sup>
</p>
<p>
where s<sub>1</sub> is the standard deviation of the first sample and s<sub>2</sub>
is the standard
deviation of the second sample. Or in code:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">sd1</span> <span class="special">/</span> <span class="identifier">sd2</span><span class="special">);</span>
<span class="identifier">F</span> <span class="special">*=</span> <span class="identifier">F</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Test Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">F</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
At this point a word of caution: the F distribution is asymmetric, so we
have to be careful how we compute the tests, the following table summarises
the options available:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Hypothesis
</p>
</th>
<th>
<p>
Test
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
The null-hypothesis: there is no difference in standard deviations
(two sided test)
</p>
</td>
<td>
<p>
Reject if F &lt;= F<sub>(1-alpha/2; N1-1, N2-1)</sub> or F &gt;= F<sub>(alpha/2;
N1-1, N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: there is a difference in means (two
sided test)
</p>
</td>
<td>
<p>
Reject if F<sub>(1-alpha/2; N1-1, N2-1)</sub> &lt;= F &lt;= F<sub>(alpha/2; N1-1,
N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: Standard deviation of sample 1 is
greater than that of sample 2
</p>
</td>
<td>
<p>
Reject if F &lt; F<sub>(alpha; N1-1, N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: Standard deviation of sample 1 is
less than that of sample 2
</p>
</td>
<td>
<p>
Reject if F &gt; F<sub>(1-alpha; N1-1, N2-1)</sub>
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
Where F<sub>(1-alpha; N1-1, N2-1)</sub> is the lower critical value of the F distribution
with degrees of freedom N1-1 and N2-1, and F<sub>(alpha; N1-1, N2-1)</sub> is the upper
critical value of the F distribution with degrees of freedom N1-1 and N2-1.
</p>
<p>
The upper and lower critical values can be computed using the quantile
function:
</p>
<p>
F<sub>(1-alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span>
<span class="identifier">alpha</span><span class="special">)</span></code>
</p>
<p>
F<sub>(alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span>
<span class="identifier">alpha</span><span class="special">))</span></code>
</p>
<p>
In our example program we need both upper and lower critical values for
alpha and for alpha/2:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ucv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">));</span>
<span class="keyword">double</span> <span class="identifier">ucv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
<span class="keyword">double</span> <span class="identifier">lcv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">lcv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
The final step is to perform the comparisons given above, and print out
whether the hypothesis is rejected or not:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span>
<span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviations are unequal (two sided test) "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special">&lt;</span> <span class="identifier">F</span><span class="special">)</span> <span class="special">||</span> <span class="special">(</span><span class="identifier">lcv2</span> <span class="special">&gt;</span> <span class="identifier">F</span><span class="special">))</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation 1 is less than standard deviation 2 "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">&gt;</span> <span class="identifier">F</span><span class="special">)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation 1 is greater than standard deviation 2 "</span><span class="special">;</span>
<span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special">&lt;</span> <span class="identifier">F</span><span class="special">)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Using the ceramic strength data as an example we get the following output:
</p>
<pre class="programlisting">F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 240
Sample Standard Deviation = 65.549
Sample 2:
Number of Observations = 240
Sample Standard Deviation = 61.854
Test Statistic = 1.123
CDF of test statistic: = 8.148e-001
Upper Critical Value at alpha: = 1.238e+000
Upper Critical Value at alpha/2: = 1.289e+000
Lower Critical Value at alpha: = 8.080e-001
Lower Critical Value at alpha/2: = 7.756e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 REJECTED
</pre>
<p>
In this case we are unable to reject the null-hypothesis, and must instead
reject the alternative hypothesis.
</p>
<p>
By contrast let's see what happens when we use some different <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm" target="_top">sample
data</a>:, once again from the NIST Engineering Statistics Handbook:
A new procedure to assemble a device is introduced and tested for possible
improvement in time of assembly. The question being addressed is whether
the standard deviation of the new assembly process (sample 2) is better
(i.e., smaller) than the standard deviation for the old assembly process
(sample 1).
</p>
<pre class="programlisting">____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 11.00000
Sample Standard Deviation = 4.90820
Sample 2:
Number of Observations = 9.00000
Sample Standard Deviation = 2.58740
Test Statistic = 3.59847
CDF of test statistic: = 9.589e-001
Upper Critical Value at alpha: = 3.347e+000
Upper Critical Value at alpha/2: = 4.295e+000
Lower Critical Value at alpha: = 3.256e-001
Lower Critical Value at alpha/2: = 2.594e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 ACCEPTED
</pre>
<p>
In this case we take our null hypothesis as "standard deviation 1
is less than or equal to standard deviation 2", since this represents
the "no change" situation. So we want to compare the upper critical
value at <span class="emphasis"><em>alpha</em></span> (a one sided test) with the test statistic,
and since 3.35 &lt; 3.6 this hypothesis must be rejected. We therefore
conclude that there is a change for the better in our standard deviation.
</p>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<dt><span class="section"><a href="find_eg/find_location_eg.html">Find
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg"></a><a class="link" href="find_location_eg.html" title="Find Location (Mean) Example">Find
Location (Mean) Example</a>
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<p>
First we need some includes to access the normal distribution, the algorithms
to find location (and some std output of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for cauchy_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cauchy</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_location</span><span class="special">;</span> <span class="comment">// for mean</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_scale</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_scale</span><span class="special">;</span> <span class="comment">// for standard devation</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">;</span> <span class="comment">// Needed if you want to use the complement version.</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
</pre>
<p>
For this example, we will use the standard normal distribution, with
mean (location) zero and standard deviation (scale) unity. This is also
the default for this implementation.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">N01</span><span class="special">;</span> <span class="comment">// Default 'standard' normal distribution with zero mean and</span>
<span class="keyword">double</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> <span class="comment">// normal default standard deviation is 1.</span>
</pre>
<p>
Suppose we want to find a different normal distribution whose mean is
shifted so that only fraction p (here 0.001 or 0.1%) are below a certain
chosen limit (here -2, two standard deviations).
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="number">2.</span><span class="special">;</span> <span class="comment">// z to give prob p</span>
<span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.001</span><span class="special">;</span> <span class="comment">// only 0.1% below z</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">N01</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">N01</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Note: uses complement.</span>
</pre>
<pre class="programlisting">Normal distribution with mean = 0, standard deviation 1, has fraction &lt;= -2, p = 0.0227501
Normal distribution with mean = 0, standard deviation 1, has fraction &gt; -2, p = 0.97725
</pre>
<p>
We can now use ''find_location'' to give a new offset mean.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"offset location (mean) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
that outputs:
</p>
<pre class="programlisting">offset location (mean) = 1.09023
</pre>
<p>
showing that we need to shift the mean just over one standard deviation
from its previous value of zero.
</p>
<p>
Then we can check that we have achieved our objective by constructing
a new distribution with the offset mean (but same standard deviation):
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">np001pc</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">);</span> <span class="comment">// Same standard_deviation (scale) but with mean (location) shifted.</span>
</pre>
<p>
And re-calculating the fraction below our chosen limit.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
<span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">np001pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
<span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">np001pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Normal distribution with mean = 1.09023 has fraction &lt;= -2, p = 0.001
Normal distribution with mean = 1.09023 has fraction &gt; -2, p = 0.999
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg.controlling_error_handling_from_"></a></span><a class="link" href="find_location_eg.html#math_toolkit.stat_tut.weg.find_eg.find_location_eg.controlling_error_handling_from_">Controlling
Error Handling from find_location</a>
</h5>
<p>
We can also control the policy for handling various errors. For example,
we can define a new (possibly unwise) policy to ignore domain errors
('bad' arguments).
</p>
<p>
Unless we are using the boost::math namespace, we will need:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">;</span>
</pre>
<p>
Using a typedef is often convenient, especially if it is re-used, although
it is not required, as the various examples below show.
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">ignore_domain_policy</span><span class="special">;</span>
<span class="comment">// find_location with new policy, using typedef.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">ignore_domain_policy</span><span class="special">());</span>
<span class="comment">// Default policy policy&lt;&gt;, needs "using boost::math::policies::policy;"</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;&gt;());</span>
<span class="comment">// Default policy, fully specified.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;());</span>
<span class="comment">// A new policy, ignoring domain errors, without using a typedef.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;());</span>
</pre>
<p>
If we want to use a probability that is the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>
of our probability, we should not even think of writing <code class="computeroutput"><span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">)</span></code>, but use the complement version, see
<a class="link" href="../../overview/complements.html#why_complements">why complements?</a>.
</p>
<pre class="programlisting"><span class="identifier">z</span> <span class="special">=</span> <span class="number">2.</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span> <span class="comment">// = 1 - p; // complement.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">));</span>
<span class="identifier">normal</span> <span class="identifier">np95pc</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">);</span> <span class="comment">// Same standard_deviation (scale) but with mean(location) shifted</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">np95pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">np95pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
See <a href="../../../../../../example/find_location_example.cpp" target="_top">find_location_example.cpp</a>
for full source code: the program output looks like this:
</p>
<pre class="programlisting">Example: Find location (mean).
Normal distribution with mean = 0, standard deviation 1, has fraction &lt;= -2, p = 0.0227501
Normal distribution with mean = 0, standard deviation 1, has fraction &gt; -2, p = 0.97725
offset location (mean) = 1.09023
Normal distribution with mean = 1.09023 has fraction &lt;= -2, p = 0.001
Normal distribution with mean = 1.09023 has fraction &gt; -2, p = 0.999
Normal distribution with mean = 0.355146 has fraction &lt;= 2 = 0.95
Normal distribution with mean = 0.355146 has fraction &gt; 2 = 0.05
</pre>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg"></a><a class="link" href="find_mean_and_sd_eg.html" title="Find mean and standard deviation example">Find
mean and standard deviation example</a>
</h5></div></div></div>
<p>
First we need some includes to access the normal distribution, the algorithms
to find location and scale (and some std output of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for cauchy_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cauchy</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_location</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_scale</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_scale</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">stdexcept</span><span class="special">&gt;</span>
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_find_location_and_find_sca"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_find_location_and_find_sca">Using
find_location and find_scale to meet dispensing and measurement specifications</a>
</h5>
<p>
Consider an example from K Krishnamoorthy, Handbook of Statistical Distributions
with Applications, ISBN 1-58488-635-8, (2006) p 126, example 10.3.7.
</p>
<p>
"A machine is set to pack 3 kg of ground beef per pack. Over a long
period of time it is found that the average packed was 3 kg with a standard
deviation of 0.1 kg. Assume the packing is normally distributed."
</p>
<p>
We start by constructing a normal distribution with the given parameters:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">3.</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="keyword">double</span> <span class="identifier">standard_deviation</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="identifier">normal</span> <span class="identifier">packs</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
</pre>
<p>
We can then find the fraction (or %) of packages that weigh more than
3.1 kg.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">max_weight</span> <span class="special">=</span> <span class="number">3.1</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Percentage of packs &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">max_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">max_weight</span><span class="special">))</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X &gt; 3.1)</span>
</pre>
<p>
We might want to ensure that 95% of packs are over a minimum weight specification,
then we want the value of the mean such that P(X &lt; 2.9) = 0.05.
</p>
<p>
Using the mean of 3 kg, we can estimate the fraction of packs that fail
to meet the specification of 2.9 kg.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">minimum_weight</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// fraction of packs &lt;= 2.9 with a mean of 3 is 0.841345</span>
</pre>
<p>
This is 0.84 - more than the target fraction of 0.95. If we want 95%
to be over the minimum weight, what should we set the mean weight to
be?
</p>
<p>
Using the KK StatCalc program supplied with the book and the method given
on page 126 gives 3.06449.
</p>
<p>
We can confirm this by constructing a new distribution which we call
'xpacks' with a safety margin mean of 3.06449 thus:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">over_mean</span> <span class="special">=</span> <span class="number">3.06449</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">xpacks</span><span class="special">(</span><span class="identifier">over_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">xpacks</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">xpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// fraction of packs &gt;= 2.9 with a mean of 3.06449 is 0.950005</span>
</pre>
<p>
Using this Math Toolkit, we can calculate the required mean directly
thus:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">under_fraction</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// so 95% are above the minimum weight mean - sd = 2.9</span>
<span class="keyword">double</span> <span class="identifier">low_limit</span> <span class="special">=</span> <span class="identifier">standard_deviation</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">offset</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">-</span> <span class="identifier">low_limit</span> <span class="special">-</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">nominal_mean</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">+</span> <span class="identifier">offset</span><span class="special">;</span>
<span class="comment">// mean + (mean - low_limit - quantile(packs, under_fraction));</span>
<span class="identifier">normal</span> <span class="identifier">nominal_packs</span><span class="special">(</span><span class="identifier">nominal_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nominal_mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">nominal_packs</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Setting the packer to 3.06449 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
</pre>
<p>
This calculation is generalized as the free function called <code class="computeroutput"><span class="identifier">find_location</span></code>, see <a class="link" href="../../../dist_ref/dist_algorithms.html" title="Distribution Algorithms">algorithms</a>.
</p>
<p>
To use this we will need to
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_location</span><span class="special">;</span>
</pre>
<p>
and then use find_location function to find safe_mean, &amp; construct
a new normal distribution called 'goodpacks'.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">safe_mean</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">normal</span> <span class="identifier">good_packs</span><span class="special">(</span><span class="identifier">safe_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
</pre>
<p>
with the same confirmation as before:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nominal_mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">good_packs</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Setting the packer to 3.06449 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h1"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_cauchy_lorentz_instead_of_"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_cauchy_lorentz_instead_of_">Using
Cauchy-Lorentz instead of normal distribution</a>
</h5>
<p>
After examining the weight distribution of a large number of packs, we
might decide that, after all, the assumption of a normal distribution
is not really justified. We might find that the fit is better to a <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy Distribution</a>.
This distribution has wider 'wings', so that whereas most of the values
are closer to the mean than the normal, there are also more values than
'normal' that lie further from the mean than the normal.
</p>
<p>
This might happen because a larger than normal lump of meat is either
included or excluded.
</p>
<p>
We first create a <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy
Distribution</a> with the original mean and standard deviation, and
estimate the fraction that lie below our minimum weight specification.
</p>
<pre class="programlisting"><span class="identifier">cauchy</span> <span class="identifier">cpacks</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">cpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Cauchy Setting the packer to 3 will mean that fraction of packs &gt;= 2.9 is 0.75</span>
</pre>
<p>
Note that far fewer of the packs meet the specification, only 75% instead
of 95%. Now we can repeat the find_location, using the cauchy distribution
as template parameter, in place of the normal used above.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">lc</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">cauchy</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_location&lt;cauchy&gt;(minimum_weight, over fraction, standard_deviation); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">lc</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// find_location&lt;cauchy&gt;(minimum_weight, over fraction, packs.standard_deviation()); 3.53138</span>
</pre>
<p>
Note that the safe_mean setting needs to be much higher, 3.53138 instead
of 3.06449, so we will make rather less profit.
</p>
<p>
And again confirm that the fraction meeting specification is as expected.
</p>
<pre class="programlisting"><span class="identifier">cauchy</span> <span class="identifier">goodcpacks</span><span class="special">(</span><span class="identifier">lc</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">lc</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Cauchy Setting the packer to 3.53138 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
</pre>
<p>
Finally we could estimate the effect of a much tighter specification,
that 99% of packs met the specification.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">cauchy</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="number">0.99</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Setting the packer to 3.13263 will mean that fraction of packs &gt;=
2.9 is 0.99, but will more than double the mean loss from 0.0644 to 0.133
kg per pack.
</p>
<p>
Of course, this calculation is not limited to packs of meat, it applies
to dispensing anything, and it also applies to a 'virtual' material like
any measurement.
</p>
<p>
The only caveat is that the calculation assumes that the standard deviation
(scale) is known with a reasonably low uncertainty, something that is
not so easy to ensure in practice. And that the distribution is well
defined, <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal
Distribution</a> or <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy
Distribution</a>, or some other.
</p>
<p>
If one is simply dispensing a very large number of packs, then it may
be feasible to measure the weight of hundreds or thousands of packs.
With a healthy 'degrees of freedom', the confidence intervals for the
standard deviation are not too wide, typically about + and - 10% for
hundreds of observations.
</p>
<p>
For other applications, where it is more difficult or expensive to make
many observations, the confidence intervals are depressingly wide.
</p>
<p>
See <a class="link" href="../cs_eg/chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation">Confidence
Intervals on the standard deviation</a> for a worked example <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>
of estimating these intervals.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h2"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.changing_the_scale_or_standard_d"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.changing_the_scale_or_standard_d">Changing
the scale or standard deviation</a>
</h5>
<p>
Alternatively, we could invest in a better (more precise) packer (or
measuring device) with a lower standard deviation, or scale.
</p>
<p>
This might cost more, but would reduce the amount we have to 'give away'
in order to meet the specification.
</p>
<p>
To estimate how much better (how much smaller standard deviation) it
would have to be, we need to get the 5% quantile to be located at the
under_weight limit, 2.9
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// wanted p th quantile.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1
</p>
<p>
With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551
kg, a little below our target of 2.9 kg. So we know that the standard
deviation is going to have to be smaller.
</p>
<p>
Let's start by guessing that it (now 0.1) needs to be halved, to a standard
deviation of 0.05 kg.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack05</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Quantile of 0.05 = 2.91776, mean = 3, sd = 0.05</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.97725</span>
</pre>
<p>
So 0.05 was quite a good guess, but we are a little over the 2.9 target,
so the standard deviation could be a tiny bit more. So we could do some
more guessing to get closer, say by increasing standard deviation to
0.06 kg, constructing another new distribution called pack06.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack06</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.06</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Quantile of 0.05 = 2.90131, mean = 3, sd = 0.06</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95221</span>
</pre>
<p>
Now we are getting really close, but to do the job properly, we might
need to use root finding method, for example the tools provided, and
used elsewhere, in the Math Toolkit, see <a class="link" href="../../../roots/roots_noderiv.html" title="Root Finding Without Derivatives">root-finding
without derivatives</a>
</p>
<p>
But in this (normal) distribution case, we can and should be even smarter
and make a direct calculation.
</p>
<p>
Our required limit is minimum_weight = 2.9 kg, often called the random
variate z. For a standard normal distribution, then probability p = N((minimum_weight
- mean) / sd).
</p>
<p>
We want to find the standard deviation that would be required to meet
this limit, so that the p th quantile is located at z (minimum_weight).
In this case, the 0.05 (5%) quantile is at 2.9 kg pack weight, when the
mean is 3 kg, ensuring that 0.95 (95%) of packs are above the minimum
weight.
</p>
<p>
Rearranging, we can directly calculate the required standard deviation:
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">N01</span><span class="special">;</span> <span class="comment">// standard normal distribution with mean zero and unit standard deviation.</span>
<span class="identifier">p</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">qp</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">N01</span><span class="special">,</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">sd95</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">minimum_weight</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">qp</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"For the "</span><span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">"th quantile to be located at "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">", would need a standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd95</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// For the 0.05th quantile to be located at 2.9, would need a standard deviation of 0.0607957</span>
</pre>
<p>
We can now construct a new (normal) distribution pack95 for the 'better'
packer, and check that our distribution will meet the specification.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack95</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">sd95</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack95</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack95</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95</span>
</pre>
<p>
This calculation is generalized in the free function find_scale, as shown
below, giving the same standard deviation.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ss</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); 0.0607957</span>
</pre>
<p>
If we had defined an over_fraction, or percentage that must pass specification
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">over_fraction</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span>
</pre>
<p>
And (wrongly) written
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">sso</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
</pre>
<p>
With the default policy, we would get a message like
</p>
<pre class="programlisting">Message from thrown exception was:
Error in function boost::math::find_scale&lt;Dist, Policy&gt;(double, double, double, Policy):
Computed scale (-0.060795683191176959) is &lt;= 0! Was the complement intended?
</pre>
<p>
But this would return a <span class="bold"><strong>negative</strong></span> standard
deviation - obviously impossible. The probability should be 1 - over_fraction,
not over_fraction, thus:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ss1o</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss1o</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); 0.0607957</span>
</pre>
<p>
But notice that using '1 - over_fraction' - will lead to a loss of accuracy,
especially if over_fraction was close to unity. (See <a class="link" href="../../overview/complements.html#why_complements">why
complements?</a>). In this (very common) case, we should instead use
the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>,
giving the most accurate result.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ssc</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(complement(minimum_weight, over_fraction, packs.mean())); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ssc</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// find_scale&lt;normal&gt;(complement(minimum_weight, over_fraction, packs.mean())); 0.0607957</span>
</pre>
<p>
Note that our guess of 0.06 was close to the accurate value of 0.060795683191176959.
</p>
<p>
We can again confirm our prediction thus:
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack95c</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">ssc</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack95c</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack95c</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95</span>
</pre>
<p>
Notice that these two deceptively simple questions:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Do we over-fill to make sure we meet a minimum specification (or
under-fill to avoid an overdose)?
</li></ul></div>
<p>
and/or
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Do we measure better?
</li></ul></div>
<p>
are actually extremely common.
</p>
<p>
The weight of beef might be replaced by a measurement of more or less
anything, from drug tablet content, Apollo landing rocket firing, X-ray
treatment doses...
</p>
<p>
The scale can be variation in dispensing or uncertainty in measurement.
</p>
<p>
See <a href="../../../../../../example/find_mean_and_sd_normal.cpp" target="_top">find_mean_and_sd_normal.cpp</a>
for full source code &amp; appended program output.
</p>
</div>
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.find_eg.find_scale_eg"></a><a class="link" href="find_scale_eg.html" title="Find Scale (Standard Deviation) Example">Find
Scale (Standard Deviation) Example</a>
</h5></div></div></div>
<p>
First we need some includes to access the <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal
Distribution</a>, the algorithms to find scale (and some std output
of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_scale</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">find_scale</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">;</span> <span class="comment">// Needed if you want to use the complement version.</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span> <span class="comment">// Needed to specify the error handling policy.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
</pre>
<p>
For this example, we will use the standard <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal
Distribution</a>, with location (mean) zero and standard deviation
(scale) unity. Conveniently, this is also the default for this implementation's
constructor.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">N01</span><span class="special">;</span> <span class="comment">// Default 'standard' normal distribution with zero mean</span>
<span class="keyword">double</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> <span class="comment">// and standard deviation is 1.</span>
</pre>
<p>
Suppose we want to find a different normal distribution with standard
deviation so that only fraction p (here 0.001 or 0.1%) are below a certain
chosen limit (here -2. standard deviations).
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="number">2.</span><span class="special">;</span> <span class="comment">// z to give prob p</span>
<span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.001</span><span class="special">;</span> <span class="comment">// only 0.1% below z = -2</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span> <span class="comment">// aka N01.mean()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span> <span class="comment">// aka N01.standard_deviation()</span>
<span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">N01</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">N01</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Note: uses complement.</span>
</pre>
<pre class="programlisting">Normal distribution with mean = 0 has fraction &lt;= -2, p = 0.0227501
Normal distribution with mean = 0 has fraction &gt; -2, p = 0.97725
</pre>
<p>
Noting that p = 0.02 instead of our target of 0.001, we can now use
<code class="computeroutput"><span class="identifier">find_scale</span></code> to give a
new standard deviation.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">();</span>
<span class="keyword">double</span> <span class="identifier">s</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"scale (standard deviation) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">s</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
that outputs:
</p>
<pre class="programlisting">scale (standard deviation) = 0.647201
</pre>
<p>
showing that we need to reduce the standard deviation from 1. to 0.65.
</p>
<p>
Then we can check that we have achieved our objective by constructing
a new distribution with the new standard deviation (but same zero mean):
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">np001pc</span><span class="special">(</span><span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">(),</span> <span class="identifier">s</span><span class="special">);</span>
</pre>
<p>
And re-calculating the fraction below (and above) our chosen limit.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
<span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">np001pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
<span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
<span class="special">&lt;&lt;</span> <span class="string">", p = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">np001pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Normal distribution with mean = 0 has fraction &lt;= -2, p = 0.001
Normal distribution with mean = 0 has fraction &gt; -2, p = 0.999
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.find_eg.find_scale_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find"></a></span><a class="link" href="find_scale_eg.html#math_toolkit.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find">Controlling
how Errors from find_scale are handled</a>
</h5>
<p>
We can also control the policy for handling various errors. For example,
we can define a new (possibly unwise) policy to ignore domain errors
('bad' arguments).
</p>
<p>
Unless we are using the boost::math namespace, we will need:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">;</span>
</pre>
<p>
Using a typedef is convenient, especially if it is re-used, although
it is not required, as the various examples below show.
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">ignore_domain_policy</span><span class="special">;</span>
<span class="comment">// find_scale with new policy, using typedef.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">ignore_domain_policy</span><span class="special">());</span>
<span class="comment">// Default policy policy&lt;&gt;, needs using boost::math::policies::policy;</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;&gt;());</span>
<span class="comment">// Default policy, fully specified.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;());</span>
<span class="comment">// New policy, without typedef.</span>
<span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;());</span>
</pre>
<p>
If we want to express a probability, say 0.999, that is a complement,
<code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
<span class="identifier">p</span></code> we should not even think
of writing <code class="computeroutput"><span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span>
<span class="identifier">p</span><span class="special">,</span>
<span class="identifier">l</span><span class="special">)</span></code>,
but use the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>
version (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>).
</p>
<pre class="programlisting"><span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="number">2.</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="number">0.999</span><span class="special">;</span> <span class="comment">// = 1 - p; // complement of 0.001.</span>
<span class="identifier">sd</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">l</span><span class="special">));</span>
<span class="identifier">normal</span> <span class="identifier">np95pc</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">);</span> <span class="comment">// Same standard_deviation (scale) but with mean(scale) shifted</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">np95pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">np95pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Sadly, it is all too easy to get probabilities the wrong way round, when
you may get a warning like this:
</p>
<pre class="programlisting">Message from thrown exception was:
Error in function boost::math::find_scale&lt;Dist, Policy&gt;(complement(double, double, double, Policy)):
Computed scale (-0.48043523852179076) is &lt;= 0! Was the complement intended?
</pre>
<p>
The default error handling policy is to throw an exception with this
message, but if you chose a policy to ignore the error, the (impossible)
negative scale is quietly returned.
</p>
<p>
See <a href="../../../../../../example/find_scale_example.cpp" target="_top">find_scale_example.cpp</a>
for full source code: the program output looks like this:
</p>
<pre class="programlisting">Example: Find scale (standard deviation).
Normal distribution with mean = 0, standard deviation 1, has fraction &lt;= -2, p = 0.0227501
Normal distribution with mean = 0, standard deviation 1, has fraction &gt; -2, p = 0.97725
scale (standard deviation) = 0.647201
Normal distribution with mean = 0 has fraction &lt;= -2, p = 0.001
Normal distribution with mean = 0 has fraction &gt; -2, p = 0.999
Normal distribution with mean = 0.946339 has fraction &lt;= -2 = 0.001
Normal distribution with mean = 0.946339 has fraction &gt; -2 = 0.999
</pre>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.geometric_eg"></a><a class="link" href="geometric_eg.html" title="Geometric Distribution Examples">Geometric Distribution
Examples</a>
</h4></div></div></div>
<p>
For this example, we will opt to #define two macros to control the error
and discrete handling policies. For this simple example, we want to avoid
throwing an exception (the default policy) and just return infinity. We
want to treat the distribution as if it was continuous, so we choose a
discrete_quantile policy of real, rather than the default policy integer_round_outwards.
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span>
<span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span>
</pre>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
It is vital to #include distributions etc <span class="bold"><strong>after</strong></span>
the above #defines
</p></td></tr>
</table></div>
<p>
After that we need some includes to provide easy access to the negative
binomial distribution, and we need some std library iostream, of course.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">geometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="comment">// for geometric_distribution</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">geometric_distribution</span><span class="special">;</span> <span class="comment">//</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">geometric</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pdf</span><span class="special">;</span> <span class="comment">// Probability mass function.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">;</span> <span class="comment">// Cumulative density function.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="comment">// for negative_binomial_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="comment">// for negative_binomial_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
</pre>
<p>
It is always sensible to use try and catch blocks because defaults policies
are to throw an exception if anything goes wrong.
</p>
<p>
Simple try'n'catch blocks (see below) will ensure that you get a helpful
error message instead of an abrupt (and silent) program abort.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.geometric_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice">Throwing
a dice</a>
</h5>
<p>
The Geometric distribution describes the probability (<span class="emphasis"><em>p</em></span>)
of a number of failures to get the first success in <span class="emphasis"><em>k</em></span>
Bernoulli trials. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli
trial</a> is one with only two possible outcomes, success of failure,
and <span class="emphasis"><em>p</em></span> is the probability of success).
</p>
<p>
Suppose an 'fair' 6-face dice is thrown repeatedly:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span><span class="special">/</span><span class="number">6</span><span class="special">;</span> <span class="comment">// success_fraction (p) = 0.1666</span>
<span class="comment">// (so failure_fraction is 1 - success_fraction = 5./6 = 1- 0.1666 = 0.8333)</span>
</pre>
<p>
If the dice is thrown repeatedly until the <span class="bold"><strong>first</strong></span>
time a <span class="emphasis"><em>three</em></span> appears. The probablility distribution
of the number of times it is thrown <span class="bold"><strong>not</strong></span>
getting a <span class="emphasis"><em>three</em></span> (<span class="emphasis"><em>not-a-threes</em></span>
number of failures to get a <span class="emphasis"><em>three</em></span>) is a geometric
distribution with the success_fraction = 1/6 = 0.1666&#8202;&#775;.
</p>
<p>
We therefore start by constructing a geometric distribution with the one
parameter success_fraction, the probability of success.
</p>
<pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">g6</span><span class="special">(</span><span class="identifier">success_fraction</span><span class="special">);</span> <span class="comment">// type double by default.</span>
</pre>
<p>
To confirm, we can echo the success_fraction parameter of the distribution.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"success fraction of a six-sided dice is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">g6</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
So the probability of getting a three at the first throw (zero failures)
is
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span>
</pre>
<p>
Note that the cdf and pdf are identical because the is only one throw.
If we want the probability of getting the first <span class="emphasis"><em>three</em></span>
on the 2nd throw:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span>
</pre>
<p>
If we want the probability of getting the first <span class="emphasis"><em>three</em></span>
on the 1st or 2nd throw (allowing one failure):
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"pdf(g6, 0) + pdf(g6, 1) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Or more conveniently, and more generally, we can use the Cumulative Distribution
Function CDF.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cdf(g6, 1) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span>
</pre>
<p>
If we allow many more (12) throws, the probability of getting our <span class="emphasis"><em>three</em></span>
gets very high:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cdf(g6, 12) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">12</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.9065 or 90% probability.</span>
</pre>
<p>
If we want to be much more confident, say 99%, we can estimate the number
of throws to be this sure using the inverse or quantile.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"quantile(g6, 0.99) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span>
</pre>
<p>
Note that the value returned is not an integer: if you want an integer
result you should use either floor, round or ceil functions, or use the
policies mechanism.
</p>
<p>
See <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding
discrete quantiles</a>.
</p>
<p>
The geometric distribution is related to the negative binomial &#8192;&#8192; <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span>
<span class="identifier">p</span><span class="special">);</span></code>
with parameter <span class="emphasis"><em>r</em></span> = 1. So we could get the same result
using the negative binomial, but using the geometric the results will be
faster, and may be more accurate.
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span>
</pre>
<p>
We could also the complement to express the required probability as 1 -
0.99 = 0.01 (and get the same result):
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"quantile(complement(g6, 1 - p)) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span>
</pre>
<p>
Note too that Boost.Math geometric distribution is implemented as a continuous
function. Unlike other implementations (for example R) it <span class="bold"><strong>uses</strong></span>
the number of failures as a <span class="bold"><strong>real</strong></span> parameter,
not as an integer. If you want this integer behaviour, you may need to
enforce this by rounding the parameter you pass, probably rounding down,
to the nearest integer. For example, R returns the success fraction probability
for all values of failures from 0 to 0.999999 thus:
</p>
<pre class="programlisting">&#8192;&#8192; R&gt; formatC(pgeom(0.0001,0.5, FALSE), digits=17) " 0.5"
</pre>
<p>
So in Boost.Math the equivalent is
</p>
<pre class="programlisting"> <span class="identifier">geometric</span> <span class="identifier">g05</span><span class="special">(</span><span class="number">0.5</span><span class="special">);</span> <span class="comment">// Probability of success = 0.5 or 50%</span>
<span class="comment">// Output all potentially significant digits for the type, here double.</span>
<span class="preprocessor">#ifdef</span> <span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</span>
<span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="number">2</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">digits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;</span> <span class="special">&gt;()</span> <span class="special">*</span> <span class="number">30103UL</span><span class="special">)</span> <span class="special">/</span> <span class="number">100000UL</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"BOOST_NO_CXX11_NUMERIC_LIMITS is defined"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#else</span>
<span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">;</span>
<span class="preprocessor">#endif</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Show all potentially significant decimal digits std::numeric_limits&lt;double&gt;::max_digits10 = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">max_digits10</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special">);</span> <span class="comment">//</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns 0.5000346561579232, not exact 0.5.</span>
</pre>
<p>
To get the R discrete behaviour, you simply need to round with, for example,
the <code class="computeroutput"><span class="identifier">floor</span></code> function.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="identifier">floor</span><span class="special">(</span><span class="number">0.0001</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns exactly 0.5</span>
</pre>
<pre class="programlisting"><code class="computeroutput"><span class="special">&gt;</span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">0.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)</span> <span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span></code>
<code class="computeroutput"><span class="special">&gt;</span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">1</span></code>
<code class="computeroutput"><span class="special">&gt;</span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">"0.12500000000000003"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">2</span></code>
</pre>
<p>
shows that R makes an arbitrary round-up decision at about 1e7 from the
next integer above. This may be convenient in practice, and could be replicated
in C++ if desired.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.geometric_eg.h1"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_">Surveying
customers to find one with a faulty product</a>
</h5>
<p>
A company knows from warranty claims that 2% of their products will be
faulty, so the 'success_fraction' of finding a fault is 0.02. It wants
to interview a purchaser of faulty products to assess their 'user experience'.
</p>
<p>
To estimate how many customers they will probably need to contact in order
to find one who has suffered from the fault, we first construct a geometric
distribution with probability 0.02, and then chose a confidence, say 80%,
95%, or 99% to finding a customer with a fault. Finally, we probably want
to round up the result to the integer above using the <code class="computeroutput"><span class="identifier">ceil</span></code>
function. (We could also use a policy, but that is hardly worthwhile for
this simple application.)
</p>
<p>
(This also assumes that each customer only buys one product: if customers
bought more than one item, the probability of finding a customer with a
fault obviously improves.)
</p>
<pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="identifier">geometric</span> <span class="identifier">g</span><span class="special">(</span><span class="number">0.02</span><span class="special">);</span> <span class="comment">// On average, 2 in 100 products are faulty.</span>
<span class="keyword">double</span> <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span> <span class="comment">// 95% confidence.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" quantile(g, "</span> <span class="special">&lt;&lt;</span> <span class="identifier">c</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"To be "</span> <span class="special">&lt;&lt;</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span>
<span class="special">&lt;&lt;</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" customers."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 148</span>
<span class="identifier">c</span> <span class="special">=</span> <span class="number">0.99</span><span class="special">;</span> <span class="comment">// Very confident.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"To be "</span> <span class="special">&lt;&lt;</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span>
<span class="special">&lt;&lt;</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" customers."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 227</span>
<span class="identifier">c</span> <span class="special">=</span> <span class="number">0.80</span><span class="special">;</span> <span class="comment">// Only reasonably confident.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"To be "</span> <span class="special">&lt;&lt;</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span>
<span class="special">&lt;&lt;</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" customers."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 79</span>
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.geometric_eg.h2"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters">Basket
Ball Shooters</a>
</h5>
<p>
According to Wikipedia, average pro basket ball players get <a href="http://en.wikipedia.org/wiki/Free_throw" target="_top">free
throws</a> in the baskets 70 to 80 % of the time, but some get as high
as 95%, and others as low as 50%. Suppose we want to compare the probabilities
of failing to get a score only on the first or on the fifth shot? To start
we will consider the average shooter, say 75%. So we construct a geometric
distribution with success_fraction parameter 75/100 = 0.75.
</p>
<pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">geometric</span> <span class="identifier">gav</span><span class="special">(</span><span class="number">0.75</span><span class="special">);</span> <span class="comment">// Shooter averages 7.5 out of 10 in the basket.</span>
</pre>
<p>
What is probability of getting 1st try in the basket, that is with no failures?
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of score on 1st try = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.75</span>
</pre>
<p>
This is, of course, the success_fraction probability 75%. What is the probability
that the shooter only scores on the fifth shot? So there are 5-1 = 4 failures
before the first success.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.0029</span>
</pre>
<p>
Now compare this with the poor and the best players success fraction. We
need to constructing new distributions with the different success fractions,
and then get the corresponding probability density functions values:
</p>
<pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">gbest</span><span class="special">(</span><span class="number">0.95</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gbest</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5.9e-6</span>
<span class="identifier">geometric</span> <span class="identifier">gmediocre</span><span class="special">(</span><span class="number">0.50</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gmediocre</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.031</span>
</pre>
<p>
So we can see the very much smaller chance (0.000006) of 4 failures by
the best shooters, compared to the 0.03 of the mediocre.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.geometric_eg.h3"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.estimating_failures"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.estimating_failures">Estimating
failures</a>
</h5>
<p>
Of course one man's failure is an other man's success. So a fault can be
defined as a 'success'.
</p>
<p>
If a fault occurs once after 100 flights, then one might naively say that
the risk of fault is obviously 1 in 100 = 1/100, a probability of 0.01.
</p>
<p>
This is the best estimate we can make, but while it is the truth, it is
not the whole truth, for it hides the big uncertainty when estimating from
a single event. "One swallow doesn't make a summer." To show
the magnitude of the uncertainty, the geometric (or the negative binomial)
distribution can be used.
</p>
<p>
If we chose the popular 95% confidence in the limits, corresponding to
an alpha of 0.05, because we are calculating a two-sided interval, we must
divide alpha by two.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">100</span><span class="special">;</span> <span class="comment">// So frequency of occurrence is 1/100.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability is failure is "</span> <span class="special">&lt;&lt;</span> <span class="number">1</span><span class="special">/</span><span class="identifier">k</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.00025</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.037</span>
</pre>
<p>
So while we estimate the probability is 0.01, it might lie between 0.0003
and 0.04. Even if we relax our confidence to alpha = 90%, the bounds only
contract to 0.0005 and 0.03. And if we require a high confidence, they
widen to 0.00005 to 0.05.
</p>
<pre class="programlisting"><span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// 90% confidence.</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.0005</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.03</span>
<span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span> <span class="comment">// 99% confidence.</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5e-005</span>
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.052</span>
</pre>
<p>
In real life, there will usually be more than one event (fault or success),
when the negative binomial, which has the neccessary extra parameter, will
be needed.
</p>
<p>
As noted above, using a catch block is always a good idea, even if you
hope not to use it!
</p>
<pre class="programlisting"><span class="special">}</span>
<span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
<span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error,</span>
<span class="comment">// an overflow exception should never be thrown.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nMessage from thrown exception was:\n "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
For example, without a ignore domain error policy, if we asked for
</p>
<pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span></pre>
<p>
for example, we would get an unhelpful abort, but with a catch:
</p>
<pre class="programlisting">Message from thrown exception was:
Error in function boost::math::pdf(const exponential_distribution&lt;double&gt;&amp;, double):
Number of failures argument is -1, but must be &gt;= 0 !
</pre>
<p>
See full source C++ of this example at <a href="../../../../../example/geometric_examples.cpp" target="_top">geometric_examples.cpp</a>
</p>
<p>
<a class="link" href="neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See
negative_binomial confidence interval example.</a>
</p>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg"></a><a class="link" href="inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example">Inverse
Chi-Squared Distribution Bayes Example</a>
</h4></div></div></div>
<p>
The scaled-inversed-chi-squared distribution is the conjugate prior distribution
for the variance (&#963;<sup>2</sup>) parameter of a normal distribution with known expectation
(&#956;). As such it has widespread application in Bayesian statistics:
</p>
<p>
In <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
inference</a>, the strength of belief into certain parameter values
is itself described through a distribution. Parameters hence become themselves
modelled and interpreted as random variables.
</p>
<p>
In this worked example, we perform such a Bayesian analysis by using the
scaled-inverse-chi-squared distribution as prior and posterior distribution
for the variance parameter of a normal distribution.
</p>
<p>
For more general information on Bayesian type of analyses, see:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Andrew Gelman, John B. Carlin, Hal E. Stern, Donald B. Rubin, Bayesian
Data Analysis, 2003, ISBN 978-1439840955.
</li>
<li class="listitem">
Jim Albert, Bayesian Compution with R, Springer, 2009, ISBN 978-0387922973.
</li>
</ul></div>
<p>
(As the scaled-inversed-chi-squared is another parameterization of the
inverse-gamma distribution, this example could also have used the inverse-gamma
distribution).
</p>
<p>
Consider precision machines which produce balls for a high-quality ball
bearing. Ideally each ball should have a diameter of precisely 3000 &#956;m (3
mm). Assume that machines generally produce balls of that size on average
(mean), but individual balls can vary slightly in either direction following
(approximately) a normal distribution. Depending on various production
conditions (e.g. raw material used for balls, workplace temperature and
humidity, maintenance frequency and quality) some machines produce balls
tighter distributed around the target of 3000 &#956;m, while others produce balls
with a wider distribution. Therefore the variance parameter of the normal
distribution of the ball sizes varies from machine to machine. An extensive
survey by the precision machinery manufacturer, however, has shown that
most machines operate with a variance between 15 and 50, and near 25 &#956;m<sup>2</sup> on
average.
</p>
<p>
Using this information, we want to model the variance of the machines.
The variance is strictly positive, and therefore we look for a statistical
distribution with support in the positive domain of the real numbers. Given
the expectation of the normal distribution of the balls is known (3000
&#956;m), for reasons of conjugacy, it is customary practice in Bayesian statistics
to model the variance to be scaled-inverse-chi-squared distributed.
</p>
<p>
In a first step, we will try to use the survey information to model the
general knowledge about the variance parameter of machines measured by
the manufacturer. This will provide us with a generic prior distribution
that is applicable if nothing more specific is known about a particular
machine.
</p>
<p>
In a second step, we will then combine the prior-distribution information
in a Bayesian analysis with data on a specific single machine to derive
a posterior distribution for that machine.
</p>
<h6>
<a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_one_using_the_survey_inform"></a></span><a class="link" href="inverse_chi_squared_eg.html#math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_one_using_the_survey_inform">Step
one: Using the survey information.</a>
</h6>
<p>
Using the survey results, we try to find the parameter set of a scaled-inverse-chi-squared
distribution so that the properties of this distribution match the results.
Using the mathematical properties of the scaled-inverse-chi-squared distribution
as guideline, we see that that both the mean and mode of the scaled-inverse-chi-squared
distribution are approximately given by the scale parameter (s) of the
distribution. As the survey machines operated at a variance of 25 &#956;m<sup>2</sup> on
average, we hence set the scale parameter (s<sub>prior</sub>) of our prior distribution
equal to this value. Using some trial-and-error and calls to the global
quantile function, we also find that a value of 20 for the degrees-of-freedom
(&#957;<sub>prior</sub>) parameter is adequate so that most of the prior distribution mass
is located between 15 and 50 (see figure below).
</p>
<p>
We first construct our prior distribution using these values, and then
list out a few quantiles:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">priorDF</span> <span class="special">=</span> <span class="number">20.0</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">priorScale</span> <span class="special">=</span> <span class="number">25.0</span><span class="special">;</span>
<span class="identifier">inverse_chi_squared</span> <span class="identifier">prior</span><span class="special">(</span><span class="identifier">priorDF</span><span class="special">,</span> <span class="identifier">priorScale</span><span class="special">);</span>
<span class="comment">// Using an inverse_gamma distribution instead, we could equivalently write</span>
<span class="comment">// inverse_gamma prior(priorDF / 2.0, priorScale * priorDF / 2.0);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Prior distribution:"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 2.5% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 50% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 97.5% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
This produces this output:
</p>
<pre class="programlisting"><span class="identifier">Prior</span> <span class="identifier">distribution</span><span class="special">:</span>
<span class="number">2.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">14.6</span>
<span class="number">50</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">25.9</span>
<span class="number">97.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">52.1</span>
</pre>
<p>
Based on this distribution, we can now calculate the probability of having
a machine working with an unusual work precision (variance) at &lt;= 15
or &gt; 50. For this task, we use calls to the <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span></code> functions <code class="computeroutput"><span class="identifier">cdf</span></code>
and <code class="computeroutput"><span class="identifier">complement</span></code>, respectively,
and find a probability of about 0.031 (3.1%) for each case.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &lt;= 15: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">15.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &lt;= 25: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">25.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &gt; 50: "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">50.0</span><span class="special">))</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
This produces this output:
</p>
<pre class="programlisting"><span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&lt;=</span> <span class="number">15</span><span class="special">:</span> <span class="number">0.031</span>
<span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&lt;=</span> <span class="number">25</span><span class="special">:</span> <span class="number">0.458</span>
<span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&gt;</span> <span class="number">50</span><span class="special">:</span> <span class="number">0.0318</span>
</pre>
<p>
Therefore, only 3.1% of all precision machines produce balls with a variance
of 15 or less (particularly precise machines), but also only 3.2% of all
machines produce balls with a variance of as high as 50 or more (particularly
imprecise machines). Moreover, slightly more than one-half (1 - 0.458 =
54.2%) of the machines work at a variance greater than 25.
</p>
<p>
Notice here the distinction between a <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian</a>
analysis and a <a href="http://en.wikipedia.org/wiki/Frequentist_inference" target="_top">frequentist</a>
analysis: because we model the variance as random variable itself, we can
calculate and straightforwardly interpret probabilities for given parameter
values directly, while such an approach is not possible (and interpretationally
a strict <span class="emphasis"><em>must-not</em></span>) in the frequentist world.
</p>
<h6>
<a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.h1"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_2_investigate_a_single_mach"></a></span><a class="link" href="inverse_chi_squared_eg.html#math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_2_investigate_a_single_mach">Step
2: Investigate a single machine</a>
</h6>
<p>
In the second step, we investigate a single machine, which is suspected
to suffer from a major fault as the produced balls show fairly high size
variability. Based on the prior distribution of generic machinery performance
(derived above) and data on balls produced by the suspect machine, we calculate
the posterior distribution for that machine and use its properties for
guidance regarding continued machine operation or suspension.
</p>
<p>
It can be shown that if the prior distribution was chosen to be scaled-inverse-chi-square
distributed, then the posterior distribution is also scaled-inverse-chi-squared-distributed
(prior and posterior distributions are hence conjugate). For more details
regarding conjugacy and formula to derive the parameters set for the posterior
distribution see <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">Conjugate
prior</a>.
</p>
<p>
Given the prior distribution parameters and sample data (of size n), the
posterior distribution parameters are given by the two expressions:
</p>
<p>
&#8192;&#8192; &#957;<sub>posterior</sub> = &#957;<sub>prior</sub> + n
</p>
<p>
which gives the posteriorDF below, and
</p>
<p>
&#8192;&#8192; s<sub>posterior</sub> = (&#957;<sub>prior</sub>s<sub>prior</sub> + &#931;<sup>n</sup><sub>i=1</sub>(x<sub>i</sub> - &#956;)<sup>2</sup>) / (&#957;<sub>prior</sub> + n)
</p>
<p>
which after some rearrangement gives the formula for the posteriorScale
below.
</p>
<p>
Machine-specific data consist of 100 balls which were accurately measured
and show the expected mean of 3000 &#956;m and a sample variance of 55 (calculated
for a sample mean defined to be 3000 exactly). From these data, the prior
parameterization, and noting that the term &#931;<sup>n</sup><sub>i=1</sub>(x<sub>i</sub> - &#956;)<sup>2</sup> equals the sample
variance multiplied by n - 1, it follows that the posterior distribution
of the variance parameter is scaled-inverse-chi-squared distribution with
degrees-of-freedom (&#957;<sub>posterior</sub>) = 120 and scale (s<sub>posterior</sub>) = 49.54.
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">ballsSampleSize</span> <span class="special">=</span> <span class="number">100</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"balls sample size: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ballsSampleSize</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">ballsSampleVariance</span> <span class="special">=</span> <span class="number">55.0</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"balls sample variance: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ballsSampleVariance</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">posteriorDF</span> <span class="special">=</span> <span class="identifier">priorDF</span> <span class="special">+</span> <span class="identifier">ballsSampleSize</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"prior degrees-of-freedom: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">priorDF</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"posterior degrees-of-freedom: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">posteriorDF</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">posteriorScale</span> <span class="special">=</span>
<span class="special">(</span><span class="identifier">priorDF</span> <span class="special">*</span> <span class="identifier">priorScale</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">ballsSampleVariance</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">ballsSampleSize</span> <span class="special">-</span> <span class="number">1</span><span class="special">)))</span> <span class="special">/</span> <span class="identifier">posteriorDF</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"prior scale: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">priorScale</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"posterior scale: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">posteriorScale</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
An interesting feature here is that one needs only to know a summary statistics
of the sample to parameterize the posterior distribution: the 100 individual
ball measurements are irrelevant, just knowledge of the sample variance
and number of measurements is sufficient.
</p>
<p>
That produces this output:
</p>
<pre class="programlisting"><span class="identifier">balls</span> <span class="identifier">sample</span> <span class="identifier">size</span><span class="special">:</span> <span class="number">100</span>
<span class="identifier">balls</span> <span class="identifier">sample</span> <span class="identifier">variance</span><span class="special">:</span> <span class="number">55</span>
<span class="identifier">prior</span> <span class="identifier">degrees</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">freedom</span><span class="special">:</span> <span class="number">20</span>
<span class="identifier">posterior</span> <span class="identifier">degrees</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">freedom</span><span class="special">:</span> <span class="number">120</span>
<span class="identifier">prior</span> <span class="identifier">scale</span><span class="special">:</span> <span class="number">25</span>
<span class="identifier">posterior</span> <span class="identifier">scale</span><span class="special">:</span> <span class="number">49.5</span>
</pre>
<p>
To compare the generic machinery performance with our suspect machine,
we calculate again the same quantiles and probabilities as above, and find
a distribution clearly shifted to greater values (see figure).
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/prior_posterior_plot.svg" align="middle"></span>
</p>
<pre class="programlisting"><span class="identifier">inverse_chi_squared</span> <span class="identifier">posterior</span><span class="special">(</span><span class="identifier">posteriorDF</span><span class="special">,</span> <span class="identifier">posteriorScale</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Posterior distribution:"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 2.5% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 50% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" 97.5% quantile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &lt;= 15: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">15.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &lt;= 25: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">25.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" probability variance &gt; 50: "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">50.0</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
This produces this output:
</p>
<pre class="programlisting"><span class="identifier">Posterior</span> <span class="identifier">distribution</span><span class="special">:</span>
<span class="number">2.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">39.1</span>
<span class="number">50</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">49.8</span>
<span class="number">97.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">64.9</span>
<span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&lt;=</span> <span class="number">15</span><span class="special">:</span> <span class="number">2.97e-031</span>
<span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&lt;=</span> <span class="number">25</span><span class="special">:</span> <span class="number">8.85e-010</span>
<span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">&gt;</span> <span class="number">50</span><span class="special">:</span> <span class="number">0.489</span>
</pre>
<p>
Indeed, the probability that the machine works at a low variance (&lt;=
15) is almost zero, and even the probability of working at average or better
performance is negligibly small (less than one-millionth of a permille).
On the other hand, with an almost near-half probability (49%), the machine
operates in the extreme high variance range of &gt; 50 characteristic for
poorly performing machines.
</p>
<p>
Based on this information the operation of the machine is taken out of
use and serviced.
</p>
<p>
In summary, the Bayesian analysis allowed us to make exact probabilistic
statements about a parameter of interest, and hence provided us results
with straightforward interpretation.
</p>
<p>
A full sample output is:
</p>
<pre class="programlisting"> Inverse_chi_squared_distribution Bayes example:
Prior distribution:
2.5% quantile: 14.6
50% quantile: 25.9
97.5% quantile: 52.1
probability variance &lt;= 15: 0.031
probability variance &lt;= 25: 0.458
probability variance &gt; 50: 0.0318
balls sample size: 100
balls sample variance: 55
prior degrees-of-freedom: 20
posterior degrees-of-freedom: 120
prior scale: 25
posterior scale: 49.5
Posterior distribution:
2.5% quantile: 39.1
50% quantile: 49.8
97.5% quantile: 64.9
probability variance &lt;= 15: 2.97e-031
probability variance &lt;= 25: 8.85e-010
probability variance &gt; 50: 0.489
</pre>
<p>
(See also the reference documentation for the <a class="link" href="../../dist_ref/dists/inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
chi squared Distribution</a>.)
</p>
<p>
See the full source C++ of this example at <a href="../../../../../example/inverse_chi_squared_bayes_eg.cpp" target="_top">../../example/inverse_chi_squared_bayes_eg.cpp</a>
</p>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.stat_tut.weg.nag_library"></a><a class="link" href="nag_library.html" title="Comparison with C, R, FORTRAN-style Free Functions">Comparison with
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<p>
You are probably familiar with a statistics library that has free functions,
for example the classic <a href="http://nag.com/numeric/CL/CLdescription.asp" target="_top">NAG
C library</a> and matching <a href="http://nag.com/numeric/FL/FLdescription.asp" target="_top">NAG
FORTRAN Library</a>, <a href="http://office.microsoft.com/en-us/excel/HP052090051033.aspx" target="_top">Microsoft
Excel BINOMDIST(number_s,trials,probability_s,cumulative)</a>, <a href="http://www.r-project.org/" target="_top">R</a>, <a href="http://www.ptc.com/products/mathcad/mathcad14/mathcad_func_chart.htm" target="_top">MathCAD
pbinom</a> and many others.
</p>
<p>
If so, you may find 'Distributions as Objects' unfamiliar, if not alien.
</p>
<p>
However, <span class="bold"><strong>do not panic</strong></span>, both definition
and usage are not really very different.
</p>
<p>
A very simple example of generating the same values as the <a href="http://nag.com/numeric/CL/CLdescription.asp" target="_top">NAG
C library</a> for the binomial distribution follows. (If you find slightly
different values, the Boost C++ version, using double or better, is very
likely to be the more accurate. Of course, accuracy is not usually a concern
for most applications of this function).
</p>
<p>
The <a href="http://www.nag.co.uk/numeric/cl/manual/pdf/G01/g01bjc.pdf" target="_top">NAG
function specification</a> is
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">nag_binomial_dist</span><span class="special">(</span><span class="identifier">Integer</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">Integer</span> <span class="identifier">k</span><span class="special">,</span>
<span class="keyword">double</span> <span class="special">*</span><span class="identifier">plek</span><span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">pgtk</span><span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">peqk</span><span class="special">,</span> <span class="identifier">NagError</span> <span class="special">*</span><span class="identifier">fail</span><span class="special">)</span>
</pre>
<p>
and is called
</p>
<pre class="programlisting"><span class="identifier">g01bjc</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">plek</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">pgtk</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">peqk</span><span class="special">,</span> <span class="identifier">NAGERR_DEFAULT</span><span class="special">);</span>
</pre>
<p>
The equivalent using this Boost C++ library is:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span> <span class="comment">// Using declaration avoids very long names.</span>
<span class="identifier">binomial</span> <span class="identifier">my_dist</span><span class="special">(</span><span class="number">4</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// c.f. NAG n = 4, p = 0.5</span>
</pre>
<p>
and values can be output thus:
</p>
<pre class="programlisting"><span class="identifier">cout</span>
<span class="special">&lt;&lt;</span> <span class="identifier">my_dist</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="comment">// Echo the NAG input n = 4 trials.</span>
<span class="special">&lt;&lt;</span> <span class="identifier">my_dist</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="comment">// Echo the NAG input p = 0.5</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="comment">// NAG plek with k = 2</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="number">2</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="comment">// NAG pgtk with k = 2</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// NAG peqk with k = 2</span>
</pre>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">k</span><span class="special">)</span></code> is
equivalent to NAG library <code class="computeroutput"><span class="identifier">plek</span></code>,
lower tail probability of &lt;= k
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">k</span><span class="special">))</span></code> is
equivalent to NAG library <code class="computeroutput"><span class="identifier">pgtk</span></code>,
upper tail probability of &gt; k
</p>
<p>
<code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">k</span><span class="special">)</span></code> is
equivalent to NAG library <code class="computeroutput"><span class="identifier">peqk</span></code>,
point probability of == k
</p>
<p>
See <a href="../../../../../example/binomial_example_nag.cpp" target="_top">binomial_example_nag.cpp</a>
for details.
</p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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Confidence Limits on the Frequency of Occurrence for the Negative Binomial
Distribution</a></span></dt>
<dt><span class="section"><a href="neg_binom_eg/neg_binom_size_eg.html">Estimating
Sample Sizes for the Negative Binomial.</a></span></dt>
<dt><span class="section"><a href="neg_binom_eg/negative_binomial_example1.html">Negative
Binomial Sales Quota Example.</a></span></dt>
<dt><span class="section"><a href="neg_binom_eg/negative_binomial_example2.html">Negative
Binomial Table Printing Example.</a></span></dt>
</dl></div>
<p>
(See also the reference documentation for the <a class="link" href="../../dist_ref/dists/negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
Binomial Distribution</a>.)
</p>
</div>
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<a name="math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf"></a><a class="link" href="neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">Calculating
Confidence Limits on the Frequency of Occurrence for the Negative Binomial
Distribution</a>
</h5></div></div></div>
<p>
Imagine you have a process that follows a negative binomial distribution:
for each trial conducted, an event either occurs or does it does not,
referred to as "successes" and "failures". The frequency
with which successes occur is variously referred to as the success fraction,
success ratio, success percentage, occurrence frequency, or probability
of occurrence.
</p>
<p>
If, by experiment, you want to measure the the best estimate of success
fraction is given simply by <span class="emphasis"><em>k</em></span> / <span class="emphasis"><em>N</em></span>,
for <span class="emphasis"><em>k</em></span> successes out of <span class="emphasis"><em>N</em></span> trials.
</p>
<p>
However our confidence in that estimate will be shaped by how many trials
were conducted, and how many successes were observed. The static member
functions <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_upper_bound_on_p</span></code>
allow you to calculate the confidence intervals for your estimate of
the success fraction.
</p>
<p>
The sample program <a href="../../../../../../example/neg_binom_confidence_limits.cpp" target="_top">neg_binom_confidence_limits.cpp</a>
illustrates their use.
</p>
<p>
First we need some includes to access the negative binomial distribution
(and some basic std output of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span>
</pre>
<p>
First define a table of significance levels: these are the probabilities
that the true occurrence frequency lies outside the calculated interval:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence
that the true occurrence frequency lies <span class="bold"><strong>inside</strong></span>
the calculated interval.
</p>
<p>
We need a function to calculate and print confidence limits for an observed
frequency of occurrence that follows a negative binomial distribution.
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// trials = Total number of trials.</span>
<span class="comment">// successes = Total number of observed successes.</span>
<span class="comment">// failures = trials - successes.</span>
<span class="comment">// success_fraction = successes /trials.</span>
<span class="comment">// Print out general info:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"______________________________________________\n"</span>
<span class="string">"2-Sided Confidence Limits For Success Fraction\n"</span>
<span class="string">"______________________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of trials"</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">trials</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of successes"</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of failures"</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">trials</span> <span class="special">-</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Observed frequency of occurrence"</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="keyword">double</span><span class="special">(</span><span class="identifier">successes</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">trials</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// Print table header:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
<span class="string">"___________________________________________\n"</span>
<span class="string">"Confidence Lower Upper\n"</span>
<span class="string">" Value (%) Limit Limit\n"</span>
<span class="string">"___________________________________________\n"</span><span class="special">;</span>
</pre>
<p>
And now for the important part - the bounds themselves. For each value
of <span class="emphasis"><em>alpha</em></span>, we call <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
to obtain lower and upper bounds respectively. Note that since we are
calculating a two-sided interval, we must divide the value of alpha in
two. Had we been calculating a single-sided interval, for example: <span class="emphasis"><em>"Calculate
a lower bound so that we are P% sure that the true occurrence frequency
is greater than some value"</em></span> then we would <span class="bold"><strong>not</strong></span>
have divided by two.
</p>
<pre class="programlisting"> <span class="comment">// Now print out the upper and lower limits for the alpha table values.</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// Calculate bounds:</span>
<span class="keyword">double</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">);</span>
<span class="comment">// Print limits:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">lower</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">upper</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// void confidence_limits_on_frequency(unsigned trials, unsigned successes)</span>
</pre>
<p>
And then call confidence_limits_on_frequency with increasing numbers
of trials, but always the same success fraction 0.1, or 1 in 10.
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span> <span class="comment">// 20 trials, 2 successes, 2 in 20, = 1 in 10 = 0.1 success fraction.</span>
<span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">200</span><span class="special">,</span> <span class="number">20</span><span class="special">);</span> <span class="comment">// More trials, but same 0.1 success fraction.</span>
<span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">2000</span><span class="special">,</span> <span class="number">200</span><span class="special">);</span> <span class="comment">// Many more trials, but same 0.1 success fraction.</span>
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// int main()</span>
</pre>
<p>
Let's see some sample output for a 1 in 10 success ratio, first for a
mere 20 trials:
</p>
<pre class="programlisting">______________________________________________
2-Sided Confidence Limits For Success Fraction
______________________________________________
Number of trials = 20
Number of successes = 2
Number of failures = 18
Observed frequency of occurrence = 0.1
___________________________________________
Confidence Lower Upper
Value (%) Limit Limit
___________________________________________
50.000 0.04812 0.13554
75.000 0.03078 0.17727
90.000 0.01807 0.22637
95.000 0.01235 0.26028
99.000 0.00530 0.33111
99.900 0.00164 0.41802
99.990 0.00051 0.49202
99.999 0.00016 0.55574
</pre>
<p>
As you can see, even at the 95% confidence level the bounds (0.012 to
0.26) are really very wide, and very asymmetric about the observed value
0.1.
</p>
<p>
Compare that with the program output for a mass 2000 trials:
</p>
<pre class="programlisting">______________________________________________
2-Sided Confidence Limits For Success Fraction
______________________________________________
Number of trials = 2000
Number of successes = 200
Number of failures = 1800
Observed frequency of occurrence = 0.1
___________________________________________
Confidence Lower Upper
Value (%) Limit Limit
___________________________________________
50.000 0.09536 0.10445
75.000 0.09228 0.10776
90.000 0.08916 0.11125
95.000 0.08720 0.11352
99.000 0.08344 0.11802
99.900 0.07921 0.12336
99.990 0.07577 0.12795
99.999 0.07282 0.13206
</pre>
<p>
Now even when the confidence level is very high, the limits (at 99.999%,
0.07 to 0.13) are really quite close and nearly symmetric to the observed
value of 0.1.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg"></a><a class="link" href="neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Estimating
Sample Sizes for the Negative Binomial.</a>
</h5></div></div></div>
<p>
Imagine you have an event (let's call it a "failure" - though
we could equally well call it a success if we felt it was a 'good' event)
that you know will occur in 1 in N trials. You may want to know how many
trials you need to conduct to be P% sure of observing at least k such
failures. If the failure events follow a negative binomial distribution
(each trial either succeeds or fails) then the static member function
<code class="computeroutput"><span class="identifier">negative_binomial_distibution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_minimum_number_of_trials</span></code>
can be used to estimate the minimum number of trials required to be P%
sure of observing the desired number of failures.
</p>
<p>
The example program <a href="../../../../../../example/neg_binomial_sample_sizes.cpp" target="_top">neg_binomial_sample_sizes.cpp</a>
demonstrates its usage.
</p>
<p>
It centres around a routine that prints out a table of minimum sample
sizes (number of trials) for various probability thresholds:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">failures</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">);</span>
</pre>
<p>
First define a table of significance levels: these are the maximum acceptable
probability that <span class="emphasis"><em>failure</em></span> or fewer events will be
observed.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence
that the desired number of failures will be observed. The values range
from a very low 0.5 or 50% confidence up to an extremely high confidence
of 99.999.
</p>
<p>
Much of the rest of the program is pretty-printing, the important part
is in the calculation of minimum number of trials required for each value
of alpha using:
</p>
<pre class="programlisting"><span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
</pre>
<p>
find_minimum_number_of_trials returns a double, so <code class="computeroutput"><span class="identifier">ceil</span></code>
rounds this up to ensure we have an integral minimum number of trials.
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">failures</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// trials = number of trials</span>
<span class="comment">// failures = number of failures before achieving required success(es).</span>
<span class="comment">// p = success fraction (0 &lt;= p &lt;= 1.).</span>
<span class="comment">//</span>
<span class="comment">// Calculate how many trials we need to ensure the</span>
<span class="comment">// required number of failures DOES exceed "failures".</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="string">"Target number of failures = "</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">failures</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">", Success fraction = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Print table header:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"____________________________\n"</span>
<span class="string">"Confidence Min Number\n"</span>
<span class="string">" Value (%) Of Trials \n"</span>
<span class="string">"____________________________\n"</span><span class="special">;</span>
<span class="comment">// Now print out the data for the alpha table values.</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span> <span class="comment">// Confidence values %:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">])</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span>
<span class="comment">// find_minimum_number_of_trials</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
<span class="special">&lt;&lt;</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]))</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// void find_number_of_trials(double failures, double p)</span>
</pre>
<p>
finally we can produce some tables of minimum trials for the chosen confidence
levels:
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">500</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">500</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span>
<span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">0.9</span><span class="special">);</span>
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span> <span class="comment">// int main()</span>
</pre>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
Since we're calculating the <span class="emphasis"><em>minimum</em></span> number of
trials required, we'll err on the safe side and take the ceiling of
the result. Had we been calculating the <span class="emphasis"><em>maximum</em></span>
number of trials permitted to observe less than a certain number of
<span class="emphasis"><em>failures</em></span> then we would have taken the floor instead.
We would also have called <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
like this:
</p>
<pre class="programlisting"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]))</span>
</pre>
<p>
which would give us the largest number of trials we could conduct and
still be P% sure of observing <span class="emphasis"><em>failures or less</em></span>
failure events, when the probability of success is <span class="emphasis"><em>p</em></span>.
</p>
</td></tr>
</table></div>
<p>
We'll finish off by looking at some sample output, firstly suppose we
wish to observe at least 5 "failures" with a 50/50 (0.5) chance
of success or failure:
</p>
<pre class="programlisting">Target number of failures = 5, Success fraction = 50%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 11
75.000 14
90.000 17
95.000 18
99.000 22
99.900 27
99.990 31
99.999 36
</pre>
<p>
So 18 trials or more would yield a 95% chance that at least our 5 required
failures would be observed.
</p>
<p>
Compare that to what happens if the success ratio is 90%:
</p>
<pre class="programlisting">Target number of failures = 5.000, Success fraction = 90.000%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 57
75.000 73
90.000 91
95.000 103
99.000 127
99.900 159
99.990 189
99.999 217
</pre>
<p>
So now 103 trials are required to observe at least 5 failures with 95%
certainty.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<a name="math_toolkit.stat_tut.weg.neg_binom_eg.negative_binomial_example1"></a><a class="link" href="negative_binomial_example1.html" title="Negative Binomial Sales Quota Example.">Negative
Binomial Sales Quota Example.</a>
</h5></div></div></div>
<p>
This example program <a href="../../../../../../example/negative_binomial_example1.cpp" target="_top">negative_binomial_example1.cpp
(full source code)</a> demonstrates a simple use to find the probability
of meeting a sales quota.
</p>
<p>
Based on <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" target="_top">a
problem by Dr. Diane Evans, Professor of Mathematics at Rose-Hulman Institute
of Technology</a>.
</p>
<p>
Pat is required to sell candy bars to raise money for the 6th grade field
trip. There are thirty houses in the neighborhood, and Pat is not supposed
to return home until five candy bars have been sold. So the child goes
door to door, selling candy bars. At each house, there is a 0.4 probability
(40%) of selling one candy bar and a 0.6 probability (60%) of selling
nothing.
</p>
<p>
What is the probability mass (density) function (pdf) for selling the
last (fifth) candy bar at the nth house?
</p>
<p>
The Negative Binomial(r, p) distribution describes the probability of
k failures and r successes in k+r Bernoulli(p) trials with success on
the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli
trial</a> is one with only two possible outcomes, success of failure,
and p is the probability of success). See also <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli
distribution</a> and <a href="http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml" target="_top">Bernoulli
applications</a>.
</p>
<p>
In this example, we will deliberately produce a variety of calculations
and outputs to demonstrate the ways that the negative binomial distribution
can be implemented with this library: it is also deliberately over-commented.
</p>
<p>
First we need to #define macros to control the error and discrete handling
policies. For this simple example, we want to avoid throwing an exception
(the default policy) and just return infinity. We want to treat the distribution
as if it was continuous, so we choose a discrete_quantile policy of real,
rather than the default policy integer_round_outwards.
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span>
<span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span>
</pre>
<p>
After that we need some includes to provide easy access to the negative
binomial distribution,
</p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
It is vital to #include distributions etc <span class="bold"><strong>after</strong></span>
the above #defines
</p></td></tr>
</table></div>
<p>
and we need some std library iostream, of course.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="comment">// for negative_binomial_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pdf</span><span class="special">;</span> <span class="comment">// Probability mass function.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">;</span> <span class="comment">// Cumulative density function.</span>
<span class="keyword">using</span> <span class="special">::</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
</pre>
<p>
It is always sensible to use try and catch blocks because defaults policies
are to throw an exception if anything goes wrong.
</p>
<p>
A simple catch block (see below) will ensure that you get a helpful error
message instead of an abrupt program abort.
</p>
<pre class="programlisting"><span class="keyword">try</span>
<span class="special">{</span>
</pre>
<p>
Selling five candy bars means getting five successes, so successes r
= 5. The total number of trials (n, in this case, houses visited) this
takes is therefore = sucesses + failures or k + r = k + 5.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">sales_quota</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span> <span class="comment">// Pat's sales quota - successes (r).</span>
</pre>
<p>
At each house, there is a 0.4 probability (40%) of selling one candy
bar and a 0.6 probability (60%) of selling nothing.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">0.4</span><span class="special">;</span> <span class="comment">// success_fraction (p) - so failure_fraction is 0.6.</span>
</pre>
<p>
The Negative Binomial(r, p) distribution describes the probability of
k failures and r successes in k+r Bernoulli(p) trials with success on
the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli
trial</a> is one with only two possible outcomes, success of failure,
and p is the probability of success).
</p>
<p>
We therefore start by constructing a negative binomial distribution with
parameters sales_quota (required successes) and probability of success.
</p>
<pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="identifier">sales_quota</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> <span class="comment">// type double by default.</span>
</pre>
<p>
To confirm, display the success_fraction &amp; successes parameters of
the distribution.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Pat has a sales per house success rate of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">success_fraction</span>
<span class="special">&lt;&lt;</span> <span class="string">".\nTherefore he would, on average, sell "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">*</span> <span class="number">100</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars after trying 100 houses."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">all_houses</span> <span class="special">=</span> <span class="number">30</span><span class="special">;</span> <span class="comment">// The number of houses on the estate.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"With a success rate of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", he might expect, on average,\n"</span>
<span class="string">"to need to visit about "</span> <span class="special">&lt;&lt;</span> <span class="identifier">success_fraction</span> <span class="special">*</span> <span class="identifier">all_houses</span>
<span class="special">&lt;&lt;</span> <span class="string">" houses in order to sell all "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">successes</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" bars. "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Pat has a sales per house success rate of 0.4.
Therefore he would, on average, sell 40 bars after trying 100 houses.
With a success rate of 0.4, he might expect, on average,
to need to visit about 12 houses in order to sell all 5 bars.
</pre>
<p>
The random variable of interest is the number of houses that must be
visited to sell five candy bars, so we substitute k = n - 5 into a negative_binomial(5,
0.4) and obtain the <a class="link" href="../../../dist_ref/nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
Density Function</a> of the distribution of houses visited. Obviously,
the best possible case is that Pat makes sales on all the first five
houses.
</p>
<p>
We calculate this using the pdf function:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on the "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">5</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// == pdf(nb, 0)</span>
</pre>
<p>
Of course, he could not finish on fewer than 5 houses because he must
sell 5 candy bars. So the 5th house is the first that he could possibly
finish on.
</p>
<p>
To finish on or before the 8th house, Pat must finish at the 5th, 6th,
7th or 8th house. The probability that he will finish on <span class="bold"><strong>exactly</strong></span>
( == ) on any house is the Probability Density Function (pdf).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on the 6th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">6</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on the 7th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">7</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on the 8th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability that Pat finishes on the 6th house is 0.03072
Probability that Pat finishes on the 7th house is 0.055296
Probability that Pat finishes on the 8th house is 0.077414
</pre>
<p>
The sum of the probabilities for these houses is the Cumulative Distribution
Function (cdf). We can calculate it by adding the individual probabilities.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on or before the 8th house is sum "</span>
<span class="string">"\n"</span> <span class="special">&lt;&lt;</span> <span class="string">"pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = "</span>
<span class="comment">// Sum each of the mass/density probabilities for houses sales_quota = 5, 6, 7, &amp; 8.</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">5</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 0 failures.</span>
<span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">6</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 1 failure.</span>
<span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">7</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 2 failures.</span>
<span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 3 failures.</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367
</pre>
<p>
Or, usually better, by using the negative binomial <span class="bold"><strong>cumulative</strong></span>
distribution function.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\non or before the "</span> <span class="special">&lt;&lt;</span> <span class="number">8</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of selling his quota of 5 bars on or before the 8th house is 0.17367
</pre>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability that Pat finishes exactly on the 10th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">10</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\non or before the "</span> <span class="special">&lt;&lt;</span> <span class="number">10</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">10</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability that Pat finishes exactly on the 10th house is 0.10033
Probability of selling his quota of 5 bars on or before the 10th house is 0.3669
</pre>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes exactly on the 11th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">11</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\non or before the "</span> <span class="special">&lt;&lt;</span> <span class="number">11</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">11</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability that Pat finishes on the 11th house is 0.10033
Probability of selling his quota of 5 candy bars
on or before the 11th house is 0.46723
</pre>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes exactly on the 12th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">12</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\non or before the "</span> <span class="special">&lt;&lt;</span> <span class="number">12</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">12</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability that Pat finishes on the 12th house is 0.094596
Probability of selling his quota of 5 candy bars
on or before the 12th house is 0.56182
</pre>
<p>
Finally consider the risk of Pat not selling his quota of 5 bars even
after visiting all the houses. Calculate the probability that he <span class="emphasis"><em>will</em></span>
sell on or before the last house: Calculate the probability that he would
sell all his quota on the very last house.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that Pat finishes on the "</span> <span class="special">&lt;&lt;</span> <span class="identifier">all_houses</span>
<span class="special">&lt;&lt;</span> <span class="string">" house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Probability of selling his quota of 5 bars on the 30th house is
</p>
<pre class="programlisting">Probability that Pat finishes on the 30 house is 0.00069145
</pre>
<p>
when he'd be very unlucky indeed!
</p>
<p>
What is the probability that Pat exhausts all 30 houses in the neighborhood,
and <span class="bold"><strong>still</strong></span> doesn't sell the required
5 candy bars?
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\non or before the "</span> <span class="special">&lt;&lt;</span> <span class="identifier">all_houses</span> <span class="special">&lt;&lt;</span> <span class="string">"th house is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of selling his quota of 5 bars
on or before the 30th house is 0.99849
</pre>
<p>
/*<code class="computeroutput"><span class="identifier">So</span> <span class="identifier">the</span>
<span class="identifier">risk</span> <span class="identifier">of</span>
<span class="identifier">failing</span> <span class="identifier">even</span>
<span class="identifier">after</span> <span class="identifier">visiting</span>
<span class="identifier">all</span> <span class="identifier">the</span>
<span class="identifier">houses</span> <span class="identifier">is</span>
<span class="number">1</span> <span class="special">-</span> <span class="keyword">this</span> <span class="identifier">probability</span><span class="special">,</span> </code><code class="computeroutput"><span class="number">1</span>
<span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span>
<span class="special">-</span> <span class="identifier">sales_quota</span></code><code class="computeroutput">
<span class="identifier">But</span> <span class="keyword">using</span>
<span class="keyword">this</span> <span class="identifier">expression</span>
<span class="identifier">may</span> <span class="identifier">cause</span>
<span class="identifier">serious</span> <span class="identifier">inaccuracy</span><span class="special">,</span> <span class="identifier">so</span> <span class="identifier">it</span> <span class="identifier">would</span>
<span class="identifier">be</span> <span class="identifier">much</span>
<span class="identifier">better</span> <span class="identifier">to</span>
<span class="identifier">use</span> <span class="identifier">the</span>
<span class="identifier">complement</span> <span class="identifier">of</span>
<span class="identifier">the</span> <span class="identifier">cdf</span><span class="special">:</span> <span class="identifier">So</span> <span class="identifier">the</span> <span class="identifier">risk</span>
<span class="identifier">of</span> <span class="identifier">failing</span>
<span class="identifier">even</span> <span class="identifier">at</span><span class="special">,</span> <span class="keyword">or</span> <span class="identifier">after</span><span class="special">,</span> <span class="identifier">the</span> <span class="number">31</span><span class="identifier">th</span> <span class="special">(</span><span class="identifier">non</span><span class="special">-</span><span class="identifier">existent</span><span class="special">)</span>
<span class="identifier">houses</span> <span class="identifier">is</span>
<span class="number">1</span> <span class="special">-</span> <span class="keyword">this</span> <span class="identifier">probability</span><span class="special">,</span> </code><code class="computeroutput"><span class="number">1</span>
<span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span>
<span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span></code>` But using this expression may cause
serious inaccuracy. So it would be much better to use the __complement
of the cdf (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nProbability of failing to sell his quota of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="string">" bars\neven after visiting all "</span> <span class="special">&lt;&lt;</span> <span class="identifier">all_houses</span> <span class="special">&lt;&lt;</span> <span class="string">" houses is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of failing to sell his quota of 5 bars
even after visiting all 30 houses is 0.0015101
</pre>
<p>
We can also use the quantile (percentile), the inverse of the cdf, to
predict which house Pat will finish on. So for the 8th house:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="special">(</span><span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of meeting sales quota on or before 8th house is "</span><span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">Probability of meeting sales quota on or before 8th house is 0.174
</pre>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span>
<span class="special">&lt;&lt;</span> <span class="string">", then the finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span><span class="special">&lt;&lt;</span> <span class="string">" quantile(nb, p) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">If the confidence of meeting sales quota is 0.17367, then the finishing house is 8
</pre>
<p>
Demanding absolute certainty that all 5 will be sold, implies an infinite
number of trials. (Of course, there are only 30 houses on the estate,
so he can't ever be <span class="bold"><strong>certain</strong></span> of selling
his quota).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span>
<span class="special">&lt;&lt;</span> <span class="string">", then the finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// 1.#INF == infinity.</span>
</pre>
<pre class="programlisting">If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF
</pre>
<p>
And similarly for a few other probabilities:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special">&lt;&lt;</span> <span class="number">0.</span>
<span class="special">&lt;&lt;</span> <span class="string">", then the finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">0.</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special">&lt;&lt;</span> <span class="number">0.5</span>
<span class="special">&lt;&lt;</span> <span class="string">", then the finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special">&lt;&lt;</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.00151</span> <span class="comment">// 30 th</span>
<span class="special">&lt;&lt;</span> <span class="string">", then the finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.00151</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">If the confidence of meeting sales quota is 0, then the finishing house is 5
If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337
If the confidence of meeting sales quota is 0.99849, then the finishing house is 30
</pre>
<p>
Notice that because we chose a discrete quantile policy of real, the
result can be an 'unreal' fractional house.
</p>
<p>
If the opposite is true, we don't want to assume any confidence, then
this is tantamount to assuming that all the first sales_quota trials
will be successful sales.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If confidence of meeting quota is zero\n(we assume all houses are successful sales)"</span>
<span class="string">", then finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">If confidence of meeting quota is zero (we assume all houses are successful sales), then finishing house is 5
If confidence of meeting quota is 0, then finishing house is 5
</pre>
<p>
We can list quantiles for a few probabilities:
</p>
<pre class="programlisting"> <span class="keyword">double</span> <span class="identifier">ps</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.9</span><span class="special">,</span> <span class="number">0.95</span><span class="special">,</span> <span class="number">0.99</span><span class="special">,</span> <span class="number">0.999</span><span class="special">,</span> <span class="number">1.</span><span class="special">};</span>
<span class="comment">// Confidence as fraction = 1-alpha, as percent = 100 * (1-alpha[i]) %</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">3</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">ps</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">ps</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="identifier">i</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If confidence of meeting quota is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span>
<span class="special">&lt;&lt;</span> <span class="string">", then finishing house is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">])</span> <span class="special">+</span> <span class="identifier">sales_quota</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<pre class="programlisting">If confidence of meeting quota is 0, then finishing house is 5
If confidence of meeting quota is 0.001, then finishing house is 5
If confidence of meeting quota is 0.01, then finishing house is 5
If confidence of meeting quota is 0.05, then finishing house is 6.2
If confidence of meeting quota is 0.1, then finishing house is 7.06
If confidence of meeting quota is 0.5, then finishing house is 11.3
If confidence of meeting quota is 0.9, then finishing house is 17.8
If confidence of meeting quota is 0.95, then finishing house is 20.1
If confidence of meeting quota is 0.99, then finishing house is 24.8
If confidence of meeting quota is 0.999, then finishing house is 31.1
If confidence of meeting quota is 1, then finishing house is 1.#INF
</pre>
<p>
We could have applied a ceil function to obtain a 'worst case' integer
value for house.
</p>
<pre class="programlisting"><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">]))</span></pre>
<p>
Or, if we had used the default discrete quantile policy, integer_outside,
by omitting
</p>
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span></pre>
<p>
we would have achieved the same effect.
</p>
<p>
The real result gives some suggestion which house is most likely. For
example, compare the real and integer_outside for 95% confidence.
</p>
<pre class="programlisting">If confidence of meeting quota is 0.95, then finishing house is 20.1
If confidence of meeting quota is 0.95, then finishing house is 21
</pre>
<p>
The real value 20.1 is much closer to 20 than 21, so integer_outside
is pessimistic. We could also use integer_round_nearest policy to suggest
that 20 is more likely.
</p>
<p>
Finally, we can tabulate the probability for the last sale being exactly
on each house.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nHouse for "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sales_quota</span> <span class="special">&lt;&lt;</span> <span class="string">"th (last) sale. Probability (%)"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">sales_quota</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="identifier">all_houses</span><span class="special">+</span><span class="number">1</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">i</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">8</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">i</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting">House for 5 th (last) sale. Probability (%)
5 0.01024
6 0.04096
7 0.096256
8 0.17367
9 0.26657
10 0.3669
11 0.46723
12 0.56182
13 0.64696
14 0.72074
15 0.78272
16 0.83343
17 0.874
18 0.90583
19 0.93039
20 0.94905
21 0.96304
22 0.97342
23 0.98103
24 0.98655
25 0.99053
26 0.99337
27 0.99539
28 0.99681
29 0.9978
30 0.99849
</pre>
<p>
As noted above, using a catch block is always a good idea, even if you
do not expect to use it.
</p>
<pre class="programlisting"><span class="special">}</span>
<span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
<span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error,</span>
<span class="comment">// an overflow exception should never be thrown.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\nMessage from thrown exception was:\n "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
For example, without a ignore domain error policy, if we asked for
</p>
<pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span></pre>
<p>
for example, we would get:
</p>
<pre class="programlisting">Message from thrown exception was:
Error in function boost::math::pdf(const negative_binomial_distribution&lt;double&gt;&amp;, double):
Number of failures argument is -1, but must be &gt;= 0 !
</pre>
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<a name="math_toolkit.stat_tut.weg.neg_binom_eg.negative_binomial_example2"></a><a class="link" href="negative_binomial_example2.html" title="Negative Binomial Table Printing Example.">Negative
Binomial Table Printing Example.</a>
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<p>
Example program showing output of a table of values of cdf and pdf for
various k failures.
</p>
<pre class="programlisting"><span class="comment">// Print a table of values that can be used to plot</span>
<span class="comment">// using Excel, or some other superior graphical display tool.</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">17</span><span class="special">);</span> <span class="comment">// Use max_digits10 precision, the maximum available for a reference table.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">showpoint</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// include trailing zeros.</span>
<span class="comment">// This is a maximum possible precision for the type (here double) to suit a reference table.</span>
<span class="keyword">int</span> <span class="identifier">maxk</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;(</span><span class="number">2.</span> <span class="special">*</span> <span class="identifier">mynbdist</span><span class="special">.</span><span class="identifier">successes</span><span class="special">()</span> <span class="special">/</span> <span class="identifier">mynbdist</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">());</span>
<span class="comment">// This maxk shows most of the range of interest, probability about 0.0001 to 0.999.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="string">" k pdf cdf"</span><span class="string">"\n"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">k</span> <span class="special">&lt;</span> <span class="identifier">maxk</span><span class="special">;</span> <span class="identifier">k</span><span class="special">++)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">17</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">showpoint</span>
<span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">k</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">mynbdist</span><span class="special">,</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">k</span><span class="special">))</span>
<span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">mynbdist</span><span class="special">,</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">k</span><span class="special">))</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<pre class="programlisting"><span class="identifier">k</span> <span class="identifier">pdf</span> <span class="identifier">cdf</span>
<span class="number">0</span><span class="special">,</span> <span class="number">1.5258789062500000e-005</span> <span class="number">1.5258789062500003e-005</span>
<span class="number">1</span><span class="special">,</span> <span class="number">9.1552734375000000e-005</span> <span class="number">0.00010681152343750000</span>
<span class="number">2</span><span class="special">,</span> <span class="number">0.00030899047851562522</span> <span class="number">0.00041580200195312500</span>
<span class="number">3</span><span class="special">,</span> <span class="number">0.00077247619628906272</span> <span class="number">0.0011882781982421875</span>
<span class="number">4</span><span class="special">,</span> <span class="number">0.0015932321548461918</span> <span class="number">0.0027815103530883789</span>
<span class="number">5</span><span class="special">,</span> <span class="number">0.0028678178787231476</span> <span class="number">0.0056493282318115234</span>
<span class="number">6</span><span class="special">,</span> <span class="number">0.0046602040529251142</span> <span class="number">0.010309532284736633</span>
<span class="number">7</span><span class="special">,</span> <span class="number">0.0069903060793876605</span> <span class="number">0.017299838364124298</span>
<span class="number">8</span><span class="special">,</span> <span class="number">0.0098301179241389001</span> <span class="number">0.027129956288263202</span>
<span class="number">9</span><span class="special">,</span> <span class="number">0.013106823898851871</span> <span class="number">0.040236780187115073</span>
<span class="number">10</span><span class="special">,</span> <span class="number">0.016711200471036140</span> <span class="number">0.056947980658151209</span>
<span class="number">11</span><span class="special">,</span> <span class="number">0.020509200578089786</span> <span class="number">0.077457181236241013</span>
<span class="number">12</span><span class="special">,</span> <span class="number">0.024354675686481652</span> <span class="number">0.10181185692272265</span>
<span class="number">13</span><span class="special">,</span> <span class="number">0.028101548869017230</span> <span class="number">0.12991340579173993</span>
<span class="number">14</span><span class="special">,</span> <span class="number">0.031614242477644432</span> <span class="number">0.16152764826938440</span>
<span class="number">15</span><span class="special">,</span> <span class="number">0.034775666725408917</span> <span class="number">0.19630331499479325</span>
<span class="number">16</span><span class="special">,</span> <span class="number">0.037492515688331451</span> <span class="number">0.23379583068312471</span>
<span class="number">17</span><span class="special">,</span> <span class="number">0.039697957787645101</span> <span class="number">0.27349378847076977</span>
<span class="number">18</span><span class="special">,</span> <span class="number">0.041352039362130305</span> <span class="number">0.31484582783290005</span>
<span class="number">19</span><span class="special">,</span> <span class="number">0.042440250924291580</span> <span class="number">0.35728607875719176</span>
<span class="number">20</span><span class="special">,</span> <span class="number">0.042970754060845245</span> <span class="number">0.40025683281803687</span>
<span class="number">21</span><span class="special">,</span> <span class="number">0.042970754060845225</span> <span class="number">0.44322758687888220</span>
<span class="number">22</span><span class="special">,</span> <span class="number">0.042482450037426581</span> <span class="number">0.48571003691630876</span>
<span class="number">23</span><span class="special">,</span> <span class="number">0.041558918514873783</span> <span class="number">0.52726895543118257</span>
<span class="number">24</span><span class="special">,</span> <span class="number">0.040260202311284021</span> <span class="number">0.56752915774246648</span>
<span class="number">25</span><span class="special">,</span> <span class="number">0.038649794218832620</span> <span class="number">0.60617895196129912</span>
<span class="number">26</span><span class="special">,</span> <span class="number">0.036791631035234917</span> <span class="number">0.64297058299653398</span>
<span class="number">27</span><span class="special">,</span> <span class="number">0.034747651533277427</span> <span class="number">0.67771823452981139</span>
<span class="number">28</span><span class="special">,</span> <span class="number">0.032575923312447595</span> <span class="number">0.71029415784225891</span>
<span class="number">29</span><span class="special">,</span> <span class="number">0.030329307911589130</span> <span class="number">0.74062346575384819</span>
<span class="number">30</span><span class="special">,</span> <span class="number">0.028054609818219924</span> <span class="number">0.76867807557206813</span>
<span class="number">31</span><span class="special">,</span> <span class="number">0.025792141284492545</span> <span class="number">0.79447021685656061</span>
<span class="number">32</span><span class="special">,</span> <span class="number">0.023575629142856460</span> <span class="number">0.81804584599941710</span>
<span class="number">33</span><span class="special">,</span> <span class="number">0.021432390129869489</span> <span class="number">0.83947823612928651</span>
<span class="number">34</span><span class="special">,</span> <span class="number">0.019383705779220189</span> <span class="number">0.85886194190850684</span>
<span class="number">35</span><span class="special">,</span> <span class="number">0.017445335201298231</span> <span class="number">0.87630727710980494</span>
<span class="number">36</span><span class="special">,</span> <span class="number">0.015628112784496322</span> <span class="number">0.89193538989430121</span>
<span class="number">37</span><span class="special">,</span> <span class="number">0.013938587078064250</span> <span class="number">0.90587397697236549</span>
<span class="number">38</span><span class="special">,</span> <span class="number">0.012379666154859701</span> <span class="number">0.91825364312722524</span>
<span class="number">39</span><span class="special">,</span> <span class="number">0.010951243136991251</span> <span class="number">0.92920488626421649</span>
<span class="number">40</span><span class="special">,</span> <span class="number">0.0096507830144735539</span> <span class="number">0.93885566927869002</span>
<span class="number">41</span><span class="special">,</span> <span class="number">0.0084738582566109364</span> <span class="number">0.94732952753530097</span>
<span class="number">42</span><span class="special">,</span> <span class="number">0.0074146259745345548</span> <span class="number">0.95474415350983555</span>
<span class="number">43</span><span class="special">,</span> <span class="number">0.0064662435824429246</span> <span class="number">0.96121039709227851</span>
<span class="number">44</span><span class="special">,</span> <span class="number">0.0056212231142827853</span> <span class="number">0.96683162020656122</span>
<span class="number">45</span><span class="special">,</span> <span class="number">0.0048717266990450708</span> <span class="number">0.97170334690560634</span>
<span class="number">46</span><span class="special">,</span> <span class="number">0.0042098073105878630</span> <span class="number">0.97591315421619418</span>
<span class="number">47</span><span class="special">,</span> <span class="number">0.0036275999165703964</span> <span class="number">0.97954075413276465</span>
<span class="number">48</span><span class="special">,</span> <span class="number">0.0031174686783026818</span> <span class="number">0.98265822281106729</span>
<span class="number">49</span><span class="special">,</span> <span class="number">0.0026721160099737302</span> <span class="number">0.98533033882104104</span>
<span class="number">50</span><span class="special">,</span> <span class="number">0.0022846591885275322</span> <span class="number">0.98761499800956853</span>
<span class="number">51</span><span class="special">,</span> <span class="number">0.0019486798960970148</span> <span class="number">0.98956367790566557</span>
<span class="number">52</span><span class="special">,</span> <span class="number">0.0016582516423517923</span> <span class="number">0.99122192954801736</span>
<span class="number">53</span><span class="special">,</span> <span class="number">0.0014079495076571762</span> <span class="number">0.99262987905567457</span>
<span class="number">54</span><span class="special">,</span> <span class="number">0.0011928461106539983</span> <span class="number">0.99382272516632852</span>
<span class="number">55</span><span class="special">,</span> <span class="number">0.0010084971662802015</span> <span class="number">0.99483122233260868</span>
<span class="number">56</span><span class="special">,</span> <span class="number">0.00085091948404891532</span> <span class="number">0.99568214181665760</span>
<span class="number">57</span><span class="special">,</span> <span class="number">0.00071656377604119542</span> <span class="number">0.99639870559269883</span>
<span class="number">58</span><span class="special">,</span> <span class="number">0.00060228420831048650</span> <span class="number">0.99700098980100937</span>
<span class="number">59</span><span class="special">,</span> <span class="number">0.00050530624256557675</span> <span class="number">0.99750629604357488</span>
<span class="number">60</span><span class="special">,</span> <span class="number">0.00042319397814867202</span> <span class="number">0.99792949002172360</span>
<span class="number">61</span><span class="special">,</span> <span class="number">0.00035381791615708398</span> <span class="number">0.99828330793788067</span>
<span class="number">62</span><span class="special">,</span> <span class="number">0.00029532382517950324</span> <span class="number">0.99857863176306016</span>
<span class="number">63</span><span class="special">,</span> <span class="number">0.00024610318764958566</span> <span class="number">0.99882473495070978</span>
</pre>
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(See also the reference documentation for the <a class="link" href="../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal
Distribution</a>.)
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Miscellaneous Examples of the Normal (Gaussian) Distribution</a>
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<p>
The sample program <a href="../../../../../../example/normal_misc_examples.cpp" target="_top">normal_misc_examples.cpp</a>
illustrates their use.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h0"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.traditional_tables"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.traditional_tables">Traditional
Tables</a>
</h5>
<p>
First we need some includes to access the normal distribution (and some
std output of course).
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Example: Normal distribution, Miscellaneous Applications."</span><span class="special">;</span>
<span class="keyword">try</span>
<span class="special">{</span>
<span class="special">{</span> <span class="comment">// Traditional tables and values.</span>
</pre>
<p>
Let's start by printing some traditional tables.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">step</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> <span class="comment">// in z</span>
<span class="keyword">double</span> <span class="identifier">range</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="comment">// min and max z = -range to +range.</span>
<span class="keyword">int</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="number">17</span><span class="special">;</span> <span class="comment">// traditional tables are only computed to much lower precision.</span>
<span class="comment">// but std::numeric_limits&lt;double&gt;::max_digits10; on new Standard Libraries gives</span>
<span class="comment">// 17, the maximum number of digits that can possibly be significant.</span>
<span class="comment">// std::numeric_limits&lt;double&gt;::digits10; == 15 is number of guaranteed digits,</span>
<span class="comment">// the other two digits being 'noisy'.</span>
<span class="comment">// Construct a standard normal distribution s</span>
<span class="identifier">normal</span> <span class="identifier">s</span><span class="special">;</span> <span class="comment">// (default mean = zero, and standard deviation = unity)</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard normal distribution, mean = "</span><span class="special">&lt;&lt;</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
First the probability distribution function (pdf).
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability distribution function values"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" z "</span> <span class="string">" pdf "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">range</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">&lt;</span> <span class="identifier">range</span> <span class="special">+</span> <span class="identifier">step</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">+=</span> <span class="identifier">step</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">12</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">6</span><span class="special">);</span> <span class="comment">// default</span>
</pre>
<p>
And the area under the normal curve from -&#8734; up to z, the cumulative distribution
function (cdf).
</p>
<pre class="programlisting"><span class="comment">// For a standard normal distribution</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard normal mean = "</span><span class="special">&lt;&lt;</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">", standard deviation = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Integral (area under the curve) from - infinity up to z "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" z "</span> <span class="string">" cdf "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">range</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">&lt;</span> <span class="identifier">range</span> <span class="special">+</span> <span class="identifier">step</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">+=</span> <span class="identifier">step</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">12</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">6</span><span class="special">);</span> <span class="comment">// default</span>
</pre>
<p>
And all this you can do with a nanoscopic amount of work compared to
the team of <span class="bold"><strong>human computers</strong></span> toiling
with Milton Abramovitz and Irene Stegen at the US National Bureau of
Standards (now <a href="http://www.nist.gov" target="_top">NIST</a>). Starting
in 1938, their "Handbook of Mathematical Functions with Formulas,
Graphs and Mathematical Tables", was eventually published in 1964,
and has been reprinted numerous times since. (A major replacement is
planned at <a href="http://dlmf.nist.gov" target="_top">Digital Library of Mathematical
Functions</a>).
</p>
<p>
Pretty-printing a traditional 2-dimensional table is left as an exercise
for the student, but why bother now that the Math Toolkit lets you write
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="number">2.</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Area for z = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// to get the area for z.</span>
</pre>
<p>
Correspondingly, we can obtain the traditional 'critical' values for
significance levels. For the 95% confidence level, the significance level
usually called alpha, is 0.05 = 1 - 0.95 (for a one-sided test), so we
can write
</p>
<pre class="programlisting"> <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"95% of area has a z below "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.95</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// 95% of area has a z below 1.64485</span>
</pre>
<p>
and a two-sided test (a comparison between two levels, rather than a
one-sided test)
</p>
<pre class="programlisting"> <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"95% of area has a z between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="special">-</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// 95% of area has a z between 1.95996 and -1.95996</span>
</pre>
<p>
First, define a table of significance levels: these are the probabilities
that the true occurrence frequency lies outside the calculated interval.
</p>
<p>
It is convenient to have an alpha level for the probability that z lies
outside just one standard deviation. This will not be some nice neat
number like 0.05, but we can easily calculate it,
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha1</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// 0.3173105078629142</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">17</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"Significance level for z == 1 is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha1</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
and place in our array of favorite alpha values.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.3173105078629142</span><span class="special">,</span> <span class="comment">// z for 1 standard deviation.</span>
<span class="number">0.20</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Confidence value as % is (1 - alpha) * 100 (so alpha 0.05 == 95% confidence)
that the true occurrence frequency lies <span class="bold"><strong>inside</strong></span>
the calculated interval.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"level of significance (alpha)"</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"2-sided 1 -sided z(alpha) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Use quantile(complement(s, alpha[i]/2)) to avoid potential loss of accuracy from quantile(s, 1 - alpha[i]/2)</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Notice the distinction between one-sided (also called one-tailed) where
we are using a &gt; <span class="bold"><strong>or</strong></span> &lt; test (and
not both) and considering the area of the tail (integral) from z up to
+&#8734;, and a two-sided test where we are using two &gt; <span class="bold"><strong>and</strong></span>
&lt; tests, and thus considering two tails, from -&#8734; up to z low and z high
up to +&#8734;.
</p>
<p>
So the 2-sided values alpha[i] are calculated using alpha[i]/2.
</p>
<p>
If we consider a simple example of alpha = 0.05, then for a two-sided
test, the lower tail area from -&#8734; up to -1.96 is 0.025 (alpha/2) and the
upper tail area from +z up to +1.96 is also 0.025 (alpha/2), and the
area between -1.96 up to 12.96 is alpha = 0.95. and the sum of the two
tails is 0.025 + 0.025 = 0.05,
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h1"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.standard_deviations_either_side_"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.standard_deviations_either_side_">Standard
deviations either side of the Mean</a>
</h5>
<p>
Armed with the cumulative distribution function, we can easily calculate
the easy to remember proportion of values that lie within 1, 2 and 3
standard deviations from the mean.
</p>
<pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">3</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">showpoint</span> <span class="special">&lt;&lt;</span> <span class="string">"cdf(s, s.standard_deviation()) = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">())</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// from -infinity to 1 sd</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cdf(complement(s, s.standard_deviation())) = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction 1 standard deviation within either side of mean is "</span>
<span class="special">&lt;&lt;</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction 2 standard deviations within either side of mean is "</span>
<span class="special">&lt;&lt;</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction 3 standard deviations within either side of mean is "</span>
<span class="special">&lt;&lt;</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
To a useful precision, the 1, 2 &amp; 3 percentages are 68, 95 and 99.7,
and these are worth memorising as useful 'rules of thumb', as, for example,
in <a href="http://en.wikipedia.org/wiki/Standard_deviation" target="_top">standard
deviation</a>:
</p>
<pre class="programlisting">Fraction 1 standard deviation within either side of mean is 0.683
Fraction 2 standard deviations within either side of mean is 0.954
Fraction 3 standard deviations within either side of mean is 0.997
</pre>
<p>
We could of course get some really accurate values for these <a href="http://en.wikipedia.org/wiki/Confidence_interval" target="_top">confidence
intervals</a> by using cout.precision(15);
</p>
<pre class="programlisting">Fraction 1 standard deviation within either side of mean is 0.682689492137086
Fraction 2 standard deviations within either side of mean is 0.954499736103642
Fraction 3 standard deviations within either side of mean is 0.997300203936740
</pre>
<p>
But before you get too excited about this impressive precision, don't
forget that the <span class="bold"><strong>confidence intervals of the standard
deviation</strong></span> are surprisingly wide, especially if you have estimated
the standard deviation from only a few measurements.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h2"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.some_simple_examples"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.some_simple_examples">Some
simple examples</a>
</h5>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h3"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.life_of_light_bulbs"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.life_of_light_bulbs">Life
of light bulbs</a>
</h5>
<p>
Examples from K. Krishnamoorthy, Handbook of Statistical Distributions
with Applications, ISBN 1 58488 635 8, page 125... implemented using
the Math Toolkit library.
</p>
<p>
A few very simple examples are shown here:
</p>
<pre class="programlisting"><span class="comment">// K. Krishnamoorthy, Handbook of Statistical Distributions with Applications,</span>
<span class="comment">// ISBN 1 58488 635 8, page 125, example 10.3.5</span>
</pre>
<p>
Mean lifespan of 100 W bulbs is 1100 h with standard deviation of 100
h. Assuming, perhaps with little evidence and much faith, that the distribution
is normal, we construct a normal distribution called <span class="emphasis"><em>bulbs</em></span>
with these values:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean_life</span> <span class="special">=</span> <span class="number">1100.</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">life_standard_deviation</span> <span class="special">=</span> <span class="number">100.</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">bulbs</span><span class="special">(</span><span class="identifier">mean_life</span><span class="special">,</span> <span class="identifier">life_standard_deviation</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">expected_life</span> <span class="special">=</span> <span class="number">1000.</span><span class="special">;</span>
</pre>
<p>
The we can use the Cumulative distribution function to predict fractions
(or percentages, if * 100) that will last various lifetimes.
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction of bulbs that will last at best (&lt;=) "</span> <span class="comment">// P(X &lt;= 1000)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">expected_life</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span><span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">expected_life</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction of bulbs that will last at least (&gt;) "</span> <span class="comment">// P(X &gt; 1000)</span>
<span class="special">&lt;&lt;</span> <span class="identifier">expected_life</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span><span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">expected_life</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">min_life</span> <span class="special">=</span> <span class="number">900</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">max_life</span> <span class="special">=</span> <span class="number">1200</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction of bulbs that will last between "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">min_life</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">max_life</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">max_life</span><span class="special">)</span> <span class="comment">// P(X &lt;= 1200)</span>
<span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">min_life</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X &lt;= 900)</span>
</pre>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Real-life failures are often very ab-normal, with a significant number
that 'dead-on-arrival' or suffer failure very early in their life:
the lifetime of the survivors of 'early mortality' may be well described
by the normal distribution.
</p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h4"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.how_many_onions"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.how_many_onions">How
many onions?</a>
</h5>
<p>
Weekly demand for 5 lb sacks of onions at a store is normally distributed
with mean 140 sacks and standard deviation 10.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">140.</span><span class="special">;</span> <span class="comment">// sacks per week.</span>
<span class="keyword">double</span> <span class="identifier">standard_deviation</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">sacks</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">stock</span> <span class="special">=</span> <span class="number">160.</span><span class="special">;</span> <span class="comment">// per week.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Percentage of weeks overstocked "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="identifier">stock</span><span class="special">)</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X &lt;=160)</span>
<span class="comment">// Percentage of weeks overstocked 97.7</span>
</pre>
<p>
So there will be lots of mouldy onions! So we should be able to say what
stock level will meet demand 95% of the weeks.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">stock_95</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="number">0.95</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Store should stock "</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">stock_95</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" sacks to meet 95% of demands."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
And it is easy to estimate how to meet 80% of demand, and waste even
less.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">stock_80</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="number">0.80</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Store should stock "</span> <span class="special">&lt;&lt;</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">stock_80</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" sacks to meet 8 out of 10 demands."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h5"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.packing_beef"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.packing_beef">Packing
beef</a>
</h5>
<p>
A machine is set to pack 3 kg of ground beef per pack. Over a long period
of time it is found that the average packed was 3 kg with a standard
deviation of 0.1 kg. Assuming the packing is normally distributed, we
can find the fraction (or %) of packages that weigh more than 3.1 kg.
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">3.</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="keyword">double</span> <span class="identifier">standard_deviation</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="identifier">normal</span> <span class="identifier">packs</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">max_weight</span> <span class="special">=</span> <span class="number">3.1</span><span class="special">;</span> <span class="comment">// kg</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Percentage of packs &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">max_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">max_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X &gt; 3.1)</span>
<span class="keyword">double</span> <span class="identifier">under_weight</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"fraction of packs &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// fraction of packs &lt;= 2.9 with a mean of 3 is 0.841345</span>
<span class="comment">// This is 0.84 - more than the target 0.95</span>
<span class="comment">// Want 95% to be over this weight, so what should we set the mean weight to be?</span>
<span class="comment">// KK StatCalc says:</span>
<span class="keyword">double</span> <span class="identifier">over_mean</span> <span class="special">=</span> <span class="number">3.0664</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">xpacks</span><span class="special">(</span><span class="identifier">over_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">xpacks</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">xpacks</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// fraction of packs &gt;= 2.9 with a mean of 3.06449 is 0.950005</span>
<span class="keyword">double</span> <span class="identifier">under_fraction</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// so 95% are above the minimum weight mean - sd = 2.9</span>
<span class="keyword">double</span> <span class="identifier">low_limit</span> <span class="special">=</span> <span class="identifier">standard_deviation</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">offset</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">-</span> <span class="identifier">low_limit</span> <span class="special">-</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">nominal_mean</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">+</span> <span class="identifier">offset</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">nominal_packs</span><span class="special">(</span><span class="identifier">nominal_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nominal_mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
<span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">nominal_packs</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Setting the packer to 3.06449 will mean that fraction of packs &gt;=
2.9 is 0.95.
</p>
<p>
Setting the packer to 3.13263 will mean that fraction of packs &gt;=
2.9 is 0.99, but will more than double the mean loss from 0.0644 to 0.133.
</p>
<p>
Alternatively, we could invest in a better (more precise) packer with
a lower standard deviation.
</p>
<p>
To estimate how much better (how much smaller standard deviation) it
would have to be, we need to get the 5% quantile to be located at the
under_weight limit, 2.9
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// wanted p th quantile.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">//</span>
</pre>
<p>
Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1
</p>
<p>
With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551
kg, a little below our target of 2.9 kg. So we know that the standard
deviation is going to have to be smaller.
</p>
<p>
Let's start by guessing that it (now 0.1) needs to be halved, to a standard
deviation of 0.05
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack05</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">//</span>
</pre>
<p>
Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of
0.05 is 0.9772
</p>
<p>
So 0.05 was quite a good guess, but we are a little over the 2.9 target,
so the standard deviation could be a tiny bit more. So we could do some
more guessing to get closer, say by increasing to 0.06
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack06</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.06</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of
0.06 is 0.9522
</p>
<p>
Now we are getting really close, but to do the job properly, we could
use root finding method, for example the tools provided, and used elsewhere,
in the Math Toolkit, see <a class="link" href="../../../roots/roots_noderiv.html" title="Root Finding Without Derivatives">root-finding
without derivatives</a>.
</p>
<p>
But in this normal distribution case, we could be even smarter and make
a direct calculation.
</p>
<pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">s</span><span class="special">;</span> <span class="comment">// For standard normal distribution,</span>
<span class="keyword">double</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span> <span class="comment">// Our required limit.</span>
<span class="comment">// then probability p = N((x - mean) / sd)</span>
<span class="comment">// So if we want to find the standard deviation that would be required to meet this limit,</span>
<span class="comment">// so that the p th quantile is located at x,</span>
<span class="comment">// in this case the 0.95 (95%) quantile at 2.9 kg pack weight, when the mean is 3 kg.</span>
<span class="keyword">double</span> <span class="identifier">prob</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">sd</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">qp</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.95</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"prob = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">prob</span> <span class="special">&lt;&lt;</span> <span class="string">", quantile(p) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">qp</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// p = 0.241971, quantile(p) 1.64485</span>
<span class="comment">// Rearranging, we can directly calculate the required standard deviation:</span>
<span class="keyword">double</span> <span class="identifier">sd95</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">((</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">qp</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"If we want the "</span><span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" th quantile to be located at "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">x</span> <span class="special">&lt;&lt;</span> <span class="string">", would need a standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd95</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">normal</span> <span class="identifier">pack95</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">sd95</span><span class="special">);</span> <span class="comment">// Distribution of the 'ideal better' packer.</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">under_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
<span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack95</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
<span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack95</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.0608 is 0.95</span>
</pre>
<p>
Notice that these two deceptively simple questions (do we over-fill or
measure better) are actually very common. The weight of beef might be
replaced by a measurement of more or less anything. But the calculations
rely on the accuracy of the standard deviation - something that is almost
always less good than we might wish, especially if based on a few measurements.
</p>
<h5>
<a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.h6"></a>
<span class="phrase"><a name="math_toolkit.stat_tut.weg.normal_example.normal_misc.length_of_bolts"></a></span><a class="link" href="normal_misc.html#math_toolkit.stat_tut.weg.normal_example.normal_misc.length_of_bolts">Length
of bolts</a>
</h5>
<p>
A bolt is usable if between 3.9 and 4.1 long. From a large batch of bolts,
a sample of 50 show a mean length of 3.95 with standard deviation 0.1.
Assuming a normal distribution, what proportion is usable? The true sample
mean is unknown, but we can use the sample mean and standard deviation
to find approximate solutions.
</p>
<pre class="programlisting"> <span class="identifier">normal</span> <span class="identifier">bolts</span><span class="special">(</span><span class="number">3.95</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">top</span> <span class="special">=</span> <span class="number">4.1</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">bottom</span> <span class="special">=</span> <span class="number">3.9</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction long enough [ P(X &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">top</span> <span class="special">&lt;&lt;</span> <span class="string">") ] is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction too short [ P(X &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">bottom</span> <span class="special">&lt;&lt;</span> <span class="string">") ] is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">bottom</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction OK -between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">bottom</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">top</span>
<span class="special">&lt;&lt;</span> <span class="string">"[ P(X &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">top</span> <span class="special">&lt;&lt;</span> <span class="string">") - P(X&lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">bottom</span> <span class="special">&lt;&lt;</span> <span class="string">" ) ] is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">bottom</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction too long [ P(X &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">top</span> <span class="special">&lt;&lt;</span> <span class="string">") ] is "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"95% of bolts are shorter than "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="number">0.95</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
</pre>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<dt><span class="section"><a href="st_eg/tut_mean_intervals.html">Calculating
confidence intervals on the mean with the Students-t distribution</a></span></dt>
<dt><span class="section"><a href="st_eg/tut_mean_test.html">Testing
a sample mean for difference from a "true" mean</a></span></dt>
<dt><span class="section"><a href="st_eg/tut_mean_size.html">Estimating
how large a sample size would have to become in order to give a significant
Students-t test result with a single sample test</a></span></dt>
<dt><span class="section"><a href="st_eg/two_sample_students_t.html">Comparing
the means of two samples with the Students-t test</a></span></dt>
<dt><span class="section"><a href="st_eg/paired_st.html">Comparing
two paired samples with the Student's t distribution</a></span></dt>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<a name="math_toolkit.stat_tut.weg.st_eg.paired_st"></a><a class="link" href="paired_st.html" title="Comparing two paired samples with the Student's t distribution">Comparing
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<p>
Imagine that we have a before and after reading for each item in the
sample: for example we might have measured blood pressure before and
after administration of a new drug. We can't pool the results and compare
the means before and after the change, because each patient will have
a different baseline reading. Instead we calculate the difference between
before and after measurements in each patient, and calculate the mean
and standard deviation of the differences. To test whether a significant
change has taken place, we can then test the null-hypothesis that the
true mean is zero using the same procedure we used in the single sample
cases previously discussed.
</p>
<p>
That means we can:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a class="link" href="tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution">Calculate
confidence intervals of the mean</a>. If the endpoints of the
interval differ in sign then we are unable to reject the null-hypothesis
that there is no change.
</li>
<li class="listitem">
<a class="link" href="tut_mean_test.html" title='Testing a sample mean for difference from a "true" mean'>Test
whether the true mean is zero</a>. If the result is consistent
with a true mean of zero, then we are unable to reject the null-hypothesis
that there is no change.
</li>
<li class="listitem">
<a class="link" href="tut_mean_size.html" title="Estimating how large a sample size would have to become in order to give a significant Students-t test result with a single sample test">Calculate
how many pairs of readings we would need in order to obtain a significant
result</a>.
</li>
</ul></div>
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<a name="math_toolkit.stat_tut.weg.st_eg.tut_mean_intervals"></a><a class="link" href="tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution">Calculating
confidence intervals on the mean with the Students-t distribution</a>
</h5></div></div></div>
<p>
Let's say you have a sample mean, you may wish to know what confidence
intervals you can place on that mean. Colloquially: "I want an interval
that I can be P% sure contains the true mean". (On a technical point,
note that the interval either contains the true mean or it does not:
the meaning of the confidence level is subtly different from this colloquialism.
More background information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">NIST
site</a>).
</p>
<p>
The formula for the interval can be expressed as:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial4.svg"></span>
</p>
<p>
Where, <span class="emphasis"><em>Y<sub>s</sub></em></span> is the sample mean, <span class="emphasis"><em>s</em></span>
is the sample standard deviation, <span class="emphasis"><em>N</em></span> is the sample
size, /&#945;/ is the desired significance level and <span class="emphasis"><em>t<sub>(&#945;/2,N-1)</sub></em></span>
is the upper critical value of the Students-t distribution with <span class="emphasis"><em>N-1</em></span>
degrees of freedom.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
The quantity &#945; &#160; is the maximum acceptable risk of falsely rejecting the
null-hypothesis. The smaller the value of &#945; the greater the strength
of the test.
</p>
<p>
The confidence level of the test is defined as 1 - &#945;, and often expressed
as a percentage. So for example a significance level of 0.05, is equivalent
to a 95% confidence level. Refer to <a href="http://www.itl.nist.gov/div898/handbook/prc/section1/prc14.htm" target="_top">"What
are confidence intervals?"</a> in <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a> for more information.
</p>
</td></tr>
</table></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
The usual assumptions of <a href="http://en.wikipedia.org/wiki/Independent_and_identically-distributed_random_variables" target="_top">independent
and identically distributed (i.i.d.)</a> variables and <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
distribution</a> of course apply here, as they do in other examples.
</p></td></tr>
</table></div>
<p>
From the formula, it should be clear that:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
The width of the confidence interval decreases as the sample size
increases.
</li>
<li class="listitem">
The width increases as the standard deviation increases.
</li>
<li class="listitem">
The width increases as the <span class="emphasis"><em>confidence level increases</em></span>
(0.5 towards 0.99999 - stronger).
</li>
<li class="listitem">
The width increases as the <span class="emphasis"><em>significance level decreases</em></span>
(0.5 towards 0.00000...01 - stronger).
</li>
</ul></div>
<p>
The following example code is taken from the example program <a href="../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>.
</p>
<p>
We'll begin by defining a procedure to calculate intervals for various
confidence levels; the procedure will print these out as a table:
</p>
<pre class="programlisting"><span class="comment">// Needed includes:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Bring everything into global namespace for ease of use:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">void</span> <span class="identifier">confidence_limits_on_mean</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span> <span class="comment">// Sm = Sample Mean.</span>
<span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="comment">// Sd = Sample Standard Deviation.</span>
<span class="keyword">unsigned</span> <span class="identifier">Sn</span><span class="special">)</span> <span class="comment">// Sn = Sample Size.</span>
<span class="special">{</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="comment">// Print out general info:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
<span class="string">"__________________________________\n"</span>
<span class="string">"2-Sided Confidence Limits For Mean\n"</span>
<span class="string">"__________________________________\n\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sn</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Mean"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sm</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
We'll define a table of significance/risk levels for which we'll compute
intervals:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Note that these are the complements of the confidence/probability levels:
0.5, 0.75, 0.9 .. 0.99999).
</p>
<p>
Next we'll declare the distribution object we'll need, note that the
<span class="emphasis"><em>degrees of freedom</em></span> parameter is the sample size
less one:
</p>
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">Sn</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
Most of what follows in the program is pretty printing, so let's focus
on the calculation of the interval. First we need the t-statistic, computed
using the <span class="emphasis"><em>quantile</em></span> function and our significance
level. Note that since the significance levels are the complement of
the probability, we have to wrap the arguments in a call to <span class="emphasis"><em>complement(...)</em></span>:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">T</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
</pre>
<p>
Note that alpha was divided by two, since we'll be calculating both the
upper and lower bounds: had we been interested in a single sided interval
then we would have omitted this step.
</p>
<p>
Now to complete the picture, we'll get the (one-sided) width of the interval
from the t-statistic by multiplying by the standard deviation, and dividing
by the square root of the sample size:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">T</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">double</span><span class="special">(</span><span class="identifier">Sn</span><span class="special">));</span>
</pre>
<p>
The two-sided interval is then the sample mean plus and minus this width.
</p>
<p>
And apart from some more pretty-printing that completes the procedure.
</p>
<p>
Let's take a look at some sample output, first using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm" target="_top">Heat
flow data</a> from the NIST site. The data set was collected by Bob
Zarr of NIST in January, 1990 from a heat flow meter calibration and
stability analysis. The corresponding dataplot output for this test can
be found in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">section
3.5.2</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>.
</p>
<pre class="programlisting"> __________________________________
2-Sided Confidence Limits For Mean
__________________________________
Number of Observations = 195
Mean = 9.26146
Standard Deviation = 0.02278881
___________________________________________________________________
Confidence T Interval Lower Upper
Value (%) Value Width Limit Limit
___________________________________________________________________
50.000 0.676 1.103e-003 9.26036 9.26256
75.000 1.154 1.883e-003 9.25958 9.26334
90.000 1.653 2.697e-003 9.25876 9.26416
95.000 1.972 3.219e-003 9.25824 9.26468
99.000 2.601 4.245e-003 9.25721 9.26571
99.900 3.341 5.453e-003 9.25601 9.26691
99.990 3.973 6.484e-003 9.25498 9.26794
99.999 4.537 7.404e-003 9.25406 9.26886
</pre>
<p>
As you can see the large sample size (195) and small standard deviation
(0.023) have combined to give very small intervals, indeed we can be
very confident that the true mean is 9.2.
</p>
<p>
For comparison the next example data output is taken from <span class="emphasis"><em>P.K.Hou,
O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics
for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J.
N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span> The values result
from the determination of mercury by cold-vapour atomic absorption.
</p>
<pre class="programlisting"> __________________________________
2-Sided Confidence Limits For Mean
__________________________________
Number of Observations = 3
Mean = 37.8000000
Standard Deviation = 0.9643650
___________________________________________________________________
Confidence T Interval Lower Upper
Value (%) Value Width Limit Limit
___________________________________________________________________
50.000 0.816 0.455 37.34539 38.25461
75.000 1.604 0.893 36.90717 38.69283
90.000 2.920 1.626 36.17422 39.42578
95.000 4.303 2.396 35.40438 40.19562
99.000 9.925 5.526 32.27408 43.32592
99.900 31.599 17.594 20.20639 55.39361
99.990 99.992 55.673 -17.87346 93.47346
99.999 316.225 176.067 -138.26683 213.86683
</pre>
<p>
This time the fact that there are only three measurements leads to much
wider intervals, indeed such large intervals that it's hard to be very
confident in the location of the mean.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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<a name="math_toolkit.stat_tut.weg.st_eg.tut_mean_size"></a><a class="link" href="tut_mean_size.html" title="Estimating how large a sample size would have to become in order to give a significant Students-t test result with a single sample test">Estimating
how large a sample size would have to become in order to give a significant
Students-t test result with a single sample test</a>
</h5></div></div></div>
<p>
Imagine you have conducted a Students-t test on a single sample in order
to check for systematic errors in your measurements. Imagine that the
result is borderline. At this point one might go off and collect more
data, but it might be prudent to first ask the question "How much
more?". The parameter estimators of the students_t_distribution
class can provide this information.
</p>
<p>
This section is based on the example code in <a href="../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>
and we begin by defining a procedure that will print out a table of estimated
sample sizes for various confidence levels:
</p>
<pre class="programlisting"><span class="comment">// Needed includes:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Bring everything into global namespace for ease of use:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">void</span> <span class="identifier">single_sample_find_df</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">M</span><span class="special">,</span> <span class="comment">// M = true mean.</span>
<span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span> <span class="comment">// Sm = Sample Mean.</span>
<span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">)</span> <span class="comment">// Sd = Sample Standard Deviation.</span>
<span class="special">{</span>
</pre>
<p>
Next we define a table of significance levels:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
Printing out the table of sample sizes required for various confidence
levels begins with the table header:
</p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
<span class="string">"_______________________________________________________________\n"</span>
<span class="string">"Confidence Estimated Estimated\n"</span>
<span class="string">" Value (%) Sample Size Sample Size\n"</span>
<span class="string">" (one sided test) (two sided test)\n"</span>
<span class="string">"_______________________________________________________________\n"</span><span class="special">;</span>
</pre>
<p>
And now the important part: the sample sizes required. Class <code class="computeroutput"><span class="identifier">students_t_distribution</span></code> has a static
member function <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
that will calculate how large a sample size needs to be in order to give
a definitive result.
</p>
<p>
The first argument is the difference between the means that you wish
to be able to detect, here it's the absolute value of the difference
between the sample mean, and the true mean.
</p>
<p>
Then come two probability values: alpha and beta. Alpha is the maximum
acceptable risk of rejecting the null-hypothesis when it is in fact true.
Beta is the maximum acceptable risk of failing to reject the null-hypothesis
when in fact it is false. Also note that for a two-sided test, alpha
must be divided by 2.
</p>
<p>
The final parameter of the function is the standard deviation of the
sample.
</p>
<p>
In this example, we assume that alpha and beta are the same, and call
<code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
twice: once with alpha for a one-sided test, and once with alpha/2 for
a two-sided test.
</p>
<pre class="programlisting"> <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// Confidence value:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
<span class="comment">// calculate df for single sided test:</span>
<span class="keyword">double</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
<span class="comment">// convert to sample size:</span>
<span class="keyword">double</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
<span class="comment">// Print size:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span><span class="special">;</span>
<span class="comment">// calculate df for two sided test:</span>
<span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
<span class="comment">// convert to sample size:</span>
<span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
<span class="comment">// Print size:</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
Let's now look at some sample output using data taken from <span class="emphasis"><em>P.K.Hou,
O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics
for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J.
N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span> The values result
from the determination of mercury by cold-vapour atomic absorption.
</p>
<p>
Only three measurements were made, and the Students-t test above gave
a borderline result, so this example will show us how many samples would
need to be collected:
</p>
<pre class="programlisting">_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Mean = 38.90000
Sample Mean = 37.80000
Sample Standard Deviation = 0.96437
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(one sided test) (two sided test)
_______________________________________________________________
75.000 3 4
90.000 7 9
95.000 11 13
99.000 20 22
99.900 35 37
99.990 50 53
99.999 66 68
</pre>
<p>
So in this case, many more measurements would have had to be made, for
example at the 95% level, 14 measurements in total for a two-sided test.
</p>
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.st_eg.tut_mean_test"></a><a class="link" href="tut_mean_test.html" title='Testing a sample mean for difference from a "true" mean'>Testing
a sample mean for difference from a "true" mean</a>
</h5></div></div></div>
<p>
When calibrating or comparing a scientific instrument or measurement
method of some kind, we want to be answer the question "Does an
observed sample mean differ from the "true" mean in any significant
way?". If it does, then we have evidence of a systematic difference.
This question can be answered with a Students-t test: more information
can be found <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">on
the NIST site</a>.
</p>
<p>
Of course, the assignment of "true" to one mean may be quite
arbitrary, often this is simply a "traditional" method of measurement.
</p>
<p>
The following example code is taken from the example program <a href="../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>.
</p>
<p>
We'll begin by defining a procedure to determine which of the possible
hypothesis are rejected or not-rejected at a given significance level:
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Non-statisticians might say 'not-rejected' means 'accepted', (often
of the null-hypothesis) implying, wrongly, that there really <span class="bold"><strong>IS</strong></span> no difference, but statisticans eschew this
to avoid implying that there is positive evidence of 'no difference'.
'Not-rejected' here means there is <span class="bold"><strong>no evidence</strong></span>
of difference, but there still might well be a difference. For example,
see <a href="http://en.wikipedia.org/wiki/Argument_from_ignorance" target="_top">argument
from ignorance</a> and <a href="http://www.bmj.com/cgi/content/full/311/7003/485" target="_top">Absence
of evidence does not constitute evidence of absence.</a>
</p></td></tr>
</table></div>
<pre class="programlisting"><span class="comment">// Needed includes:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Bring everything into global namespace for ease of use:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">void</span> <span class="identifier">single_sample_t_test</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">M</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">Sn</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">//</span>
<span class="comment">// M = true mean.</span>
<span class="comment">// Sm = Sample Mean.</span>
<span class="comment">// Sd = Sample Standard Deviation.</span>
<span class="comment">// Sn = Sample Size.</span>
<span class="comment">// alpha = Significance Level.</span>
</pre>
<p>
Most of the procedure is pretty-printing, so let's just focus on the
calculation, we begin by calculating the t-statistic:
</p>
<pre class="programlisting"><span class="comment">// Difference in means:</span>
<span class="keyword">double</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">Sm</span> <span class="special">-</span> <span class="identifier">M</span><span class="special">;</span>
<span class="comment">// Degrees of freedom:</span>
<span class="keyword">unsigned</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sn</span> <span class="special">-</span> <span class="number">1</span><span class="special">;</span>
<span class="comment">// t-statistic:</span>
<span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="identifier">diff</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">double</span><span class="special">(</span><span class="identifier">Sn</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">Sd</span><span class="special">;</span>
</pre>
<p>
Finally calculate the probability from the t-statistic. If we're interested
in simply whether there is a difference (either less or greater) or not,
we don't care about the sign of the t-statistic, and we take the complement
of the probability for comparison to the significance level:
</p>
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t_stat</span><span class="special">)));</span>
</pre>
<p>
The procedure then prints out the results of the various tests that can
be done, these can be summarised in the following table:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Hypothesis
</p>
</th>
<th>
<p>
Test
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span>
in means
</p>
</td>
<td>
<p>
Reject if complement of CDF for |t| &lt; significance level
/ 2:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
<span class="special">&lt;</span> <span class="identifier">alpha</span>
<span class="special">/</span> <span class="number">2</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: there <span class="bold"><strong>is
difference</strong></span> in means
</p>
</td>
<td>
<p>
Reject if complement of CDF for |t| &gt; significance level
/ 2:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
<span class="special">&gt;</span> <span class="identifier">alpha</span>
<span class="special">/</span> <span class="number">2</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: the sample mean <span class="bold"><strong>is
less</strong></span> than the true mean.
</p>
</td>
<td>
<p>
Reject if CDF of t &gt; 1 - significance level:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">t</span><span class="special">))</span>
<span class="special">&lt;</span> <span class="identifier">alpha</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: the sample mean <span class="bold"><strong>is
greater</strong></span> than the true mean.
</p>
</td>
<td>
<p>
Reject if complement of CDF of t &lt; significance level:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">t</span><span class="special">)</span>
<span class="special">&lt;</span> <span class="identifier">alpha</span></code>
</p>
</td>
</tr>
</tbody>
</table></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Notice that the comparisons are against <code class="computeroutput"><span class="identifier">alpha</span>
<span class="special">/</span> <span class="number">2</span></code>
for a two-sided test and against <code class="computeroutput"><span class="identifier">alpha</span></code>
for a one-sided test
</p></td></tr>
</table></div>
<p>
Now that we have all the parts in place, let's take a look at some sample
output, first using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm" target="_top">Heat
flow data</a> from the NIST site. The data set was collected by Bob
Zarr of NIST in January, 1990 from a heat flow meter calibration and
stability analysis. The corresponding dataplot output for this test can
be found in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">section
3.5.2</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>.
</p>
<pre class="programlisting">__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 195
Sample Mean = 9.26146
Sample Standard Deviation = 0.02279
Expected True Mean = 5.00000
Sample Mean - Expected Test Mean = 4.26146
Degrees of Freedom = 194
T Statistic = 2611.28380
Probability that difference is due to chance = 0.000e+000
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 5.000 NOT REJECTED
Mean &lt; 5.000 REJECTED
Mean &gt; 5.000 NOT REJECTED
</pre>
<p>
You will note the line that says the probability that the difference
is due to chance is zero. From a philosophical point of view, of course,
the probability can never reach zero. However, in this case the calculated
probability is smaller than the smallest representable double precision
number, hence the appearance of a zero here. Whatever its "true"
value is, we know it must be extraordinarily small, so the alternative
hypothesis - that there is a difference in means - is not rejected.
</p>
<p>
For comparison the next example data output is taken from <span class="emphasis"><em>P.K.Hou,
O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics
for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J.
N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span> The values result
from the determination of mercury by cold-vapour atomic absorption.
</p>
<pre class="programlisting">__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 3
Sample Mean = 37.80000
Sample Standard Deviation = 0.96437
Expected True Mean = 38.90000
Sample Mean - Expected Test Mean = -1.10000
Degrees of Freedom = 2
T Statistic = -1.97566
Probability that difference is due to chance = 1.869e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 38.900 REJECTED
Mean &lt; 38.900 NOT REJECTED
Mean &gt; 38.900 NOT REJECTED
</pre>
<p>
As you can see the small number of measurements (3) has led to a large
uncertainty in the location of the true mean. So even though there appears
to be a difference between the sample mean and the expected true mean,
we conclude that there is no significant difference, and are unable to
reject the null hypothesis. However, if we were to lower the bar for
acceptance down to alpha = 0.1 (a 90% confidence level) we see a different
output:
</p>
<pre class="programlisting">__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 3
Sample Mean = 37.80000
Sample Standard Deviation = 0.96437
Expected True Mean = 38.90000
Sample Mean - Expected Test Mean = -1.10000
Degrees of Freedom = 2
T Statistic = -1.97566
Probability that difference is due to chance = 1.869e-001
Results for Alternative Hypothesis and alpha = 0.1000
Alternative Hypothesis Conclusion
Mean != 38.900 REJECTED
Mean &lt; 38.900 NOT REJECTED
Mean &gt; 38.900 REJECTED
</pre>
<p>
In this case, we really have a borderline result, and more data (and/or
more accurate data), is needed for a more convincing conclusion.
</p>
</div>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.st_eg.two_sample_students_t"></a><a class="link" href="two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test">Comparing
the means of two samples with the Students-t test</a>
</h5></div></div></div>
<p>
Imagine that we have two samples, and we wish to determine whether their
means are different or not. This situation often arises when determining
whether a new process or treatment is better than an old one.
</p>
<p>
In this example, we'll be using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm" target="_top">Car
Mileage sample data</a> from the <a href="http://www.itl.nist.gov" target="_top">NIST
website</a>. The data compares miles per gallon of US cars with miles
per gallon of Japanese cars.
</p>
<p>
The sample code is in <a href="../../../../../../example/students_t_two_samples.cpp" target="_top">students_t_two_samples.cpp</a>.
</p>
<p>
There are two ways in which this test can be conducted: we can assume
that the true standard deviations of the two samples are equal or not.
If the standard deviations are assumed to be equal, then the calculation
of the t-statistic is greatly simplified, so we'll examine that case
first. In real life we should verify whether this assumption is valid
with a Chi-Squared test for equal variances.
</p>
<p>
We begin by defining a procedure that will conduct our test assuming
equal variances:
</p>
<pre class="programlisting"><span class="comment">// Needed headers:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Simplify usage:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">void</span> <span class="identifier">two_samples_t_test_equal_sd</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">Sm1</span><span class="special">,</span> <span class="comment">// Sm1 = Sample 1 Mean.</span>
<span class="keyword">double</span> <span class="identifier">Sd1</span><span class="special">,</span> <span class="comment">// Sd1 = Sample 1 Standard Deviation.</span>
<span class="keyword">unsigned</span> <span class="identifier">Sn1</span><span class="special">,</span> <span class="comment">// Sn1 = Sample 1 Size.</span>
<span class="keyword">double</span> <span class="identifier">Sm2</span><span class="special">,</span> <span class="comment">// Sm2 = Sample 2 Mean.</span>
<span class="keyword">double</span> <span class="identifier">Sd2</span><span class="special">,</span> <span class="comment">// Sd2 = Sample 2 Standard Deviation.</span>
<span class="keyword">unsigned</span> <span class="identifier">Sn2</span><span class="special">,</span> <span class="comment">// Sn2 = Sample 2 Size.</span>
<span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// alpha = Significance Level.</span>
<span class="special">{</span>
</pre>
<p>
Our procedure will begin by calculating the t-statistic, assuming equal
variances the needed formulae are:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial1.svg"></span>
</p>
<p>
where Sp is the "pooled" standard deviation of the two samples,
and <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the
two combined samples. We can now write the code to calculate the t-statistic:
</p>
<pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
<span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">2</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// Pooled variance:</span>
<span class="keyword">double</span> <span class="identifier">sp</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(((</span><span class="identifier">Sn1</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">Sn2</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">v</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Pooled Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sp</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// t-statistic:</span>
<span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sp</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
The next step is to define our distribution object, and calculate the
complement of the probability:
</p>
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t_stat</span><span class="special">)));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that difference is due to chance"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">q</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
Here we've used the absolute value of the t-statistic, because we initially
want to know simply whether there is a difference or not (a two-sided
test). However, we can also test whether the mean of the second sample
is greater or is less (one-sided test) than that of the first: all the
possible tests are summed up in the following table:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Hypothesis
</p>
</th>
<th>
<p>
Test
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span>
in means
</p>
</td>
<td>
<p>
Reject if complement of CDF for |t| &lt; significance level
/ 2:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
<span class="special">&lt;</span> <span class="identifier">alpha</span>
<span class="special">/</span> <span class="number">2</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: there is a <span class="bold"><strong>difference</strong></span>
in means
</p>
</td>
<td>
<p>
Reject if complement of CDF for |t| &gt; significance level
/ 2:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
<span class="special">&lt;</span> <span class="identifier">alpha</span>
<span class="special">/</span> <span class="number">2</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>less</strong></span>
than Sample 2 Mean.
</p>
</td>
<td>
<p>
Reject if CDF of t &gt; significance level:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">t</span><span class="special">)</span>
<span class="special">&gt;</span> <span class="identifier">alpha</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>greater</strong></span>
than Sample 2 Mean.
</p>
</td>
<td>
<p>
Reject if complement of CDF of t &gt; significance level:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
<span class="identifier">t</span><span class="special">))</span>
<span class="special">&gt;</span> <span class="identifier">alpha</span></code>
</p>
</td>
</tr>
</tbody>
</table></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
For a two-sided test we must compare against alpha / 2 and not alpha.
</p></td></tr>
</table></div>
<p>
Most of the rest of the sample program is pretty-printing, so we'll skip
over that, and take a look at the sample output for alpha=0.05 (a 95%
probability level). For comparison the dataplot output for the same data
is in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm" target="_top">section
1.3.5.3</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
e-Handbook of Statistical Methods.</a>.
</p>
<pre class="programlisting"> ________________________________________________
Student t test for two samples (equal variances)
________________________________________________
Number of Observations (Sample 1) = 249
Sample 1 Mean = 20.145
Sample 1 Standard Deviation = 6.4147
Number of Observations (Sample 2) = 79
Sample 2 Mean = 30.481
Sample 2 Standard Deviation = 6.1077
Degrees of Freedom = 326
Pooled Standard Deviation = 6.3426
T Statistic = -12.621
Probability that difference is due to chance = 5.273e-030
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean NOT REJECTED
Sample 1 Mean &lt; Sample 2 Mean NOT REJECTED
Sample 1 Mean &gt; Sample 2 Mean REJECTED
</pre>
<p>
So with a probability that the difference is due to chance of just 5.273e-030,
we can safely conclude that there is indeed a difference.
</p>
<p>
The tests on the alternative hypothesis show that we must also reject
the hypothesis that Sample 1 Mean is greater than that for Sample 2:
in this case Sample 1 represents the miles per gallon for Japanese cars,
and Sample 2 the miles per gallon for US cars, so we conclude that Japanese
cars are on average more fuel efficient.
</p>
<p>
Now that we have the simple case out of the way, let's look for a moment
at the more complex one: that the standard deviations of the two samples
are not equal. In this case the formula for the t-statistic becomes:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial2.svg"></span>
</p>
<p>
And for the combined degrees of freedom we use the <a href="http://en.wikipedia.org/wiki/Welch-Satterthwaite_equation" target="_top">Welch-Satterthwaite</a>
approximation:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial3.svg"></span>
</p>
<p>
Note that this is one of the rare situations where the degrees-of-freedom
parameter to the Student's t distribution is a real number, and not an
integer value.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Some statistical packages truncate the effective degrees of freedom
to an integer value: this may be necessary if you are relying on lookup
tables, but since our code fully supports non-integer degrees of freedom
there is no need to truncate in this case. Also note that when the
degrees of freedom is small then the Welch-Satterthwaite approximation
may be a significant source of error.
</p></td></tr>
</table></div>
<p>
Putting these formulae into code we get:
</p>
<pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
<span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
<span class="identifier">v</span> <span class="special">*=</span> <span class="identifier">v</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">t1</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span><span class="special">;</span>
<span class="identifier">t1</span> <span class="special">*=</span> <span class="identifier">t1</span><span class="special">;</span>
<span class="identifier">t1</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">Sn1</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">t2</span> <span class="special">=</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
<span class="identifier">t2</span> <span class="special">*=</span> <span class="identifier">t2</span><span class="special">;</span>
<span class="identifier">t2</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">v</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">t1</span> <span class="special">+</span> <span class="identifier">t2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// t-statistic:</span>
<span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
Thereafter the code and the tests are performed the same as before. Using
are car mileage data again, here's what the output looks like:
</p>
<pre class="programlisting"> __________________________________________________
Student t test for two samples (unequal variances)
__________________________________________________
Number of Observations (Sample 1) = 249
Sample 1 Mean = 20.145
Sample 1 Standard Deviation = 6.4147
Number of Observations (Sample 2) = 79
Sample 2 Mean = 30.481
Sample 2 Standard Deviation = 6.1077
Degrees of Freedom = 136.87
T Statistic = -12.946
Probability that difference is due to chance = 1.571e-025
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean NOT REJECTED
Sample 1 Mean &lt; Sample 2 Mean NOT REJECTED
Sample 1 Mean &gt; Sample 2 Mean REJECTED
</pre>
<p>
This time allowing the variances in the two samples to differ has yielded
a higher likelihood that the observed difference is down to chance alone
(1.571e-025 compared to 5.273e-030 when equal variances were assumed).
However, the conclusion remains the same: US cars are less fuel efficient
than Japanese models.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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