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307 lines
9.4 KiB
HTML
Executable File
307 lines
9.4 KiB
HTML
Executable File
<HTML><HEAD>
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<TITLE> Random Variate Generation Routines </TITLE>
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</HEAD><BODY>
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<H1> Random Variate Generation Routines </H1>
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<P>This module provides facilities for basic pseudo-random number generation,
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and for generation of random variates from various common distributions.
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<P>All the random generation procedures use the same underlying
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"stream" of random numbers. This stream is generated using the
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<TT>nrand48</TT> pseudo-random generation procedure found on most Unix
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systems, with the output being combined with a limited supply of real
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random numbers from the file "randfile" in the source
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directory for this software, in order to reduce the chances that the
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pseudo-random numbers aren't random enough.
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<P>A pseudo-random number stream is determined by an integer
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<I>seed</I>, which is typically set by the user for each experimental
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run (eg, to 1, 2, 3, etc. for successive runs). The state of the
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random number stream can be saved in a structure of type
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<TT>rand_state</TT>, and later restored. For example, the state could
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be saved in a file at the end of a run, and restored if it is later
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decided that the run should be continued for longer.
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<P>The methods for generating random variates from various distributions
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are mostly taken from the following reference:
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<BLOCKQUOTE>
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Devroye, L. (1986) <I>Non-Uniform Random Variate Generation</I>,
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New York: Springer-Verlag.
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</BLOCKQUOTE>
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The methods used here are not necessarily the fastest available. They were
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selected to be reasonably fast while also being easy to write.
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<P><B>Header file required</B>: <TT>rand.h</TT>
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<A NAME="get-set-sec">
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<P><HR>
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<CENTER><BIG>Setting and saving the state</BIG></CENTER>
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</A>
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<A NAME="seed"><HR><B>rand_seed</B>:
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Set the state from an integer seed.</A>
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<BLOCKQUOTE><PRE>
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void rand_seed
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( int seed; /* Seed to set state based on */
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)
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</PRE></BLOCKQUOTE>
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<P>Sets the random number generator to a state that is determined from
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the integer <TT>seed</TT>. Setting the seed the same on two occasions,
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and then calling exactly the same sequence of random generation
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procedures, should produce exactly the same sequence of random
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variates. (Note: this may not be true, however, if computers with
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different architectures are used on the two occasions.)
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<P>Sequential seeds should produce streams that are pretty much
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independent (unlike the situation for some pseudo-random number
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generators). Setting the seed randomly according to the time or day,
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or some such, without recording what it was, is <B>not</B>
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recommended, since the results are then not reproducible.
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<P><A NAME="get_state"><HR><B>rand_get_state</B>:
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Get a pointer to the current state.</A>
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<BLOCKQUOTE><PRE>
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rand_state *rand_get_state (void)
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</PRE></BLOCKQUOTE>
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Returns a pointer to the current state of the random number generator,
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which is a structure of type <TT>rand_state</TT>. The only use for this
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pointer is to use it to save a copy of the current state someplace.
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<P><A NAME="use_state"><HR><B>rand_use_state</B>:
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Set the state to use from now on.</A>
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<BLOCKQUOTE><PRE>
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void rand_use_state
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( rand_state *state /* Pointer to a previously saved state */
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)
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</PRE></BLOCKQUOTE>
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Sets the random number generator to the state pointed to by the argument.
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This state must be valid. The only way to get a valid state is by using
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<TT>rand_get_state</TT>.
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<A NAME="uniform-sec">
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<P><HR>
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<CENTER><BIG>Generating uniform random variates</BIG></CENTER>
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</A>
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<A NAME="uniform"><HR><B>rand_uniform</B>:
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Generate uniformly from [0,1).</A>
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<BLOCKQUOTE><PRE>
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double rand_uniform (void)
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</PRE></BLOCKQUOTE>
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Returns a random number that is uniformly distributed between 0 and 1,
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with a value of exactly 0 being possible, but with a value of exactly 1
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not being possible.
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<P><A NAME="uniopen"><HR><B>rand_uniopen</B>:
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Generate uniformly from (0,1).</A>
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<BLOCKQUOTE><PRE>
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double rand_uniopen (void)
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</PRE></BLOCKQUOTE>
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Returns a random number that is uniformly distributed between 0 and 1,
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with neither a value of exactly 0 nor a value of exactly 1
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being possible.
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<A NAME="discrete-sec">
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<P><HR>
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<CENTER><BIG>Generating discrete random variates</BIG></CENTER>
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</A>
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<A NAME="int"><HR><B>rand_int</B>:
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Pick an integer uniformly from <I>0</I> to <I>n-1</I>.</A>
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<BLOCKQUOTE><PRE>
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int rand_int
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( int n /* Number of integers (from 0) to pick from */
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)
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</PRE></BLOCKQUOTE>
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<P>Randomly picks an integer from the set { <TT>0, 1, ..., n-1</TT> },
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with each integer being equally probable. The probabilities may
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become somewhat unequal, however, if <TT>n</TT> is very large,
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approaching 2<SUP><SMALL>31</SMALL></SUP>.
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<P><A NAME="pickd"><HR><B>rand_pickd</B>: Pick an integer from <I>0</I>
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to <I>n-1</I> with given probabilities (as doubles).</A>
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<BLOCKQUOTE><PRE>
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int rand_pickd
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( double *p, /* Array of probabilities, of length n */
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int n /* Number of integers (from 0) to pick from */
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)
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</PRE></BLOCKQUOTE>
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<P>Randomly picks an integer from the set { <TT>0, 1, ..., n-1</TT> },
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with probabilities given by the array of double-precision numbers
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passed as the first argument. These numbers need not sum to one, but they
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must be non-negative, and must not all be zero. The actual
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probabilities used are obtained by dividing each number in this array
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by the sum of all the numbers.
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<P>If one of the probabilities is exactly zero, it is guaranteed that
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the corresponding integer will not be picked.
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<P><A NAME="pickf"><HR><B>rand_pickf</B>: Pick an integer from <I>0</I>
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to <I>n-1</I> with given probabilities (as floats).</A>
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<BLOCKQUOTE><PRE>
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int rand_pickf
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( float *p, /* Array of probabilities, of length n */
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int n /* Number of integers (from 0) to pick from */
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)
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</PRE></BLOCKQUOTE>
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<P>This procedure is the same as <A HREF="#pickd"><TT>rand_pickd</TT></A>
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except that the array giving the probabilities is an array of single-precision
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floating point numbers, rather than double-precision.
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<A NAME="poisson"><HR><B>rand_poisson</B>:
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Generate a non-negative integer from a Poisson distribution.</A>
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<BLOCKQUOTE><PRE>
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int rand_poisson
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( double lambda /* Mean of the Poisson distribution */
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)
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</PRE></BLOCKQUOTE>
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<P>Generates a non-negative integer from the Poisson distribution with mean
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<tt>lambda</tt>.
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<P><A NAME="permutation"><HR><B>rand_permutation</B>: Generate a random
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permutation of integers from <i>1</i> to <i>n</i>.</A>
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<BLOCKQUOTE><PRE>
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int rand_permutation
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( int *perm, /* Place to store the permutation */
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int n /* Number of integers to permute */
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)
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</PRE></BLOCKQUOTE>
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<P>Stores a random permutation of the integers from <i>1</i> to <i>n</i> in the
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array perm, which must be at least <i>n</i> long. All permutations are
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equally likely.
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<A NAME="continuous-sec">
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<P><HR>
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<CENTER><BIG>Generating continuous random variates</BIG></CENTER>
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</A>
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<A NAME="gaussian"><HR><B>rand_gaussian</B>:
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Generate a standard Gaussian (normal) random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_gaussian (void)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the Gaussian (normal) distribution
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with mean zero and standard deviation one, whose density function is
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proportional to exp(-<I>x<SUP><SMALL>2</SMALL></SUP></I>/2), with <I>x</I>
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being any real value.
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<A NAME="logistic"><HR><B>rand_logistic</B>:
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Generate a logistic random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_logistic (void)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the logistic distribution with
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location parameter zero and width parameter one, whose density
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function is proportional to
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exp(-<I>x</I>) / [1 + exp(-<I>x</I>)]<SUP><SMALL>2</SMALL></SUP>,
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with <I>x</I> being any real value.
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<P><A NAME="cauchy"><HR><B>rand_cauchy</B>:
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Generate a Cauchy random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_cauchy (void)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the Cauchy distribution with centre
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at zero and width one, whose density function is proportional to
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1 / (1+<I>x<SUP><SMALL>2</SMALL></SUP></I>), with <I>x</I> being any real value.
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<P><A NAME="gamma"><HR><B>rand_gamma</B>:
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Generate a gamma-distributed random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_gamma
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( double A /* Shape parameter */
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)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the gamma distribution with
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positive shape parameter <I>A</I>, whose density function is proportional
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to <I>x<SUP><SMALL>A-1</SMALL></SUP></I> exp(-<I>x</I>), with <I>x</I>
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being any positive real. This procedure will never return a value that is
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exactly zero.
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<P><A NAME="exp"><HR><B>rand_exp</B>:
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Generate an exponentially-distributed random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_exponential (void)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the exponential distribution with
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mean one, whose density function is exp(-<I>x</I>), with <I>x</I> being
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any positive real. This procedure will never return a value that is
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exactly zero.
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<P>Note: This is a common special case of the gamma distribution.
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<P><A NAME="beta"><HR><B>rand_beta</B>:
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Generate a beta-distributed random variate.</A>
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<BLOCKQUOTE><PRE>
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double rand_beta
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( double A, /* Parameters of distribution */
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double B
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)
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</PRE></BLOCKQUOTE>
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<P>Generates a random value drawn from the beta distribution with positive
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parameters <I>A</I> and <I>B</I>, whose density function is proportional
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to <I>x<SUP><SMALL>A-1</SMALL></SUP>(1-x)<SUP><SMALL>B-1</SMALL></SUP></I>,
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with <I>x</I> being any real in the interval (0,1). This procedure
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will never return a value that is exactly zero or exactly one.
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<HR>
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