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109 lines
3.5 KiB
Plaintext
109 lines
3.5 KiB
Plaintext
[section:exp_dist Exponential Distribution]
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``#include <boost/math/distributions/exponential.hpp>``
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class exponential_distribution;
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typedef exponential_distribution<> exponential;
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template <class RealType, class ``__Policy``>
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class exponential_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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exponential_distribution(RealType lambda = 1);
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RealType lambda()const;
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};
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The [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution]
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is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution]
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with PDF:
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[equation exponential_dist_ref1]
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It is often used to model the time between independent
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events that happen at a constant average rate.
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The following graph shows how the distribution changes for different
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values of the rate parameter lambda:
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[graph exponential_pdf]
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[h4 Member Functions]
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exponential_distribution(RealType lambda = 1);
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Constructs an
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[@http://en.wikipedia.org/wiki/Exponential_distribution Exponential distribution]
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with parameter /lambda/.
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Lambda is defined as the reciprocal of the scale parameter.
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Requires lambda > 0, otherwise calls __domain_error.
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RealType lambda()const;
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Accessor function returns the lambda parameter of the 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 \[0, +[infin]\].
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[h4 Accuracy]
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The exponential distribution is implemented in terms of the
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standard library functions `exp`, `log`, `log1p` and `expm1`
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and as such should have very low error rates.
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[h4 Implementation]
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In the following table [lambda] is the parameter lambda of the distribution,
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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 = [lambda] * exp(-[lambda] * x) ]]
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[[cdf][Using the relation: p = 1 - exp(-x * [lambda]) = -expm1(-x * [lambda]) ]]
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[[cdf complement][Using the relation: q = exp(-x * [lambda]) ]]
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[[quantile][Using the relation: x = -log(1-p) / [lambda] = -log1p(-p) / [lambda]]]
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[[quantile from the complement][Using the relation: x = -log(q) / [lambda]]]
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[[mean][1/[lambda]]]
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[[standard deviation][1/[lambda]]]
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[[mode][0]]
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[[skewness][2]]
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[[kurtosis][9]]
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[[kurtosis excess][6]]
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]
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[h4 references]
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* [@http://mathworld.wolfram.com/ExponentialDistribution.html Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Web Resource]
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* [@http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html Wolfram Mathematica calculator]
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* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST Exploratory Data Analysis]
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* [@http://en.wikipedia.org/wiki/Exponential_distribution Wikipedia Exponential distribution]
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(See also the reference documentation for the related __extreme_distrib.)
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*
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[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
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Samuel Kotz & Saralees Nadarajah]
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discuss the relationship of the types of extreme value distributions.
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[endsect][/section:exp_dist Exponential]
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[/ exponential.qbk
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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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