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135 lines
4.6 KiB
Plaintext
135 lines
4.6 KiB
Plaintext
[section:uniform_dist Uniform Distribution]
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``#include <boost/math/distributions/uniform.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class uniform_distribution;
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typedef uniform_distribution<> uniform;
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template <class RealType, class ``__Policy``>
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class uniform_distribution
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{
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public:
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typedef RealType value_type;
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uniform_distribution(RealType lower = 0, RealType upper = 1); // Constructor.
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: m_lower(lower), m_upper(upper) // Default is standard uniform distribution.
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// Accessor functions.
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RealType lower()const;
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RealType upper()const;
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}; // class uniform_distribution
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}} // namespaces
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The uniform distribution, also known as a rectangular distribution,
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is a probability distribution that has constant probability.
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The [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 continuous uniform distribution]
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is a distribution with the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
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f(x) =
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* 1 / (upper - lower) for lower < x < upper
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* zero for x < lower or x > upper
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and in this implementation:
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* 1 / (upper - lower) for x = lower or x = upper
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The choice of x = lower or x = upper is made because statistical use of this distribution judged is most likely:
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the method of maximum likelihood uses this definition.
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There is also a [@http://en.wikipedia.org/wiki/Discrete_uniform_distribution *discrete* uniform distribution].
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Parameters lower and upper can be any finite value.
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The [@http://en.wikipedia.org/wiki/Random_variate random variate]
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x must also be finite, and is supported lower <= x <= upper.
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The lower parameter is also called the
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[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm location parameter],
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[@http://en.wikipedia.org/wiki/Location_parameter that is where the origin of a plot will lie],
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and (upper - lower) is also called the [@http://en.wikipedia.org/wiki/Scale_parameter scale parameter].
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The following graph illustrates how the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
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varies with the shape parameter:
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[graph uniform_pdf]
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Likewise for the CDF:
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[graph uniform_cdf]
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[h4 Member Functions]
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uniform_distribution(RealType lower = 0, RealType upper = 1);
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Constructs a [@http://en.wikipedia.org/wiki/uniform_distribution
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uniform distribution] with lower /lower/ (a) and upper /upper/ (b).
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Requires that the /lower/ and /upper/ parameters are both finite;
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otherwise if infinity or NaN then calls __domain_error.
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RealType lower()const;
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Returns the /lower/ parameter of this distribution.
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RealType upper()const;
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Returns the /upper/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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The domain of the random variable is any finite value,
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but the supported range is only /lower/ <= x <= /upper/.
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[h4 Accuracy]
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The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two.
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[h4 Implementation]
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In the following table a is the /lower/ parameter of the distribution,
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b is the /upper/ parameter,
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/x/ is the random variate, /p/ is the probability and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = 0 for x < a, 1 / (b - a) for a <= x <= b, 0 for x > b ]]
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[[cdf][Using the relation: cdf = 0 for x < a, (x - a) / (b - a) for a <= x <= b, 1 for x > b]]
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[[cdf complement][Using the relation: q = 1 - p, (b - x) / (b - a) ]]
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[[quantile][Using the relation: x = p * (b - a) + a; ]]
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[[quantile from the complement][x = -q * (b - a) + b ]]
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[[mean][(a + b) / 2 ]]
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[[variance][(b - a) [super 2] / 12 ]]
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[[mode][any value in \[a, b\] but a is chosen. (Would NaN be better?) ]]
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[[skewness][0]]
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[[kurtosis excess][-6/5 = -1.2 exactly. (kurtosis - 3)]]
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[[kurtosis][9/5]]
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]
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[h4 References]
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* [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 Wikpedia continuous uniform distribution]
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* [@http://mathworld.wolfram.com/UniformDistribution.html Weisstein, Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource.]
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* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm]
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[endsect][/section:uniform_dist Uniform]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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