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111 lines
3.6 KiB
Plaintext
111 lines
3.6 KiB
Plaintext
[section:poisson_dist Poisson Distribution]
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``#include <boost/math/distributions/poisson.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 poisson_distribution;
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typedef poisson_distribution<> poisson;
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template <class RealType, class ``__Policy``>
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class poisson_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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BOOST_MATH_GPU_ENABLED poisson_distribution(RealType mean = 1); // Constructor.
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BOOST_MATH_GPU_ENABLED RealType mean()const; // Accessor.
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}
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}} // namespaces boost::math
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The [@http://en.wikipedia.org/wiki/Poisson_distribution Poisson distribution]
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is a well-known statistical discrete distribution.
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It expresses the probability of a number of events
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(or failures, arrivals, occurrences ...)
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occurring in a fixed period of time,
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provided these events occur with a known mean rate [lambda]
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(events/time), and are independent of the time since the last event.
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The distribution was discovered by Sim[eacute]on-Denis Poisson (1781 to 1840).
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It has the Probability Mass Function:
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[equation poisson_ref1]
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for k events, with an expected number of events [lambda].
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The following graph illustrates how the PDF varies with the parameter [lambda]:
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[graph poisson_pdf_1]
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[discrete_quantile_warning Poisson]
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[h4 Member Functions]
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BOOST_MATH_GPU_ENABLED poisson_distribution(RealType mean = 1);
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Constructs a poisson distribution with mean /mean/.
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BOOST_MATH_GPU_ENABLED RealType mean()const;
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Returns the /mean/ 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] that are generic to all
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distributions are supported: __usual_accessors.
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For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
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be run on both host and device.
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The domain of the random variable is \[0, [infin]\].
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In this distribution the implementation of `logpdf` is specialized
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to improve numerical accuracy.
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[h4 Accuracy]
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The Poisson distribution is implemented in terms of the
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incomplete gamma functions __gamma_p and __gamma_q
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and as such should have low error rates: but refer to the documentation
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of those functions for more information.
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The quantile and its complement use the inverse gamma functions
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and are therefore probably slightly less accurate: this is because the
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inverse gamma functions are implemented using an iterative method with a
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lower tolerance to avoid excessive computation.
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[h4 Implementation]
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In the following table [lambda] is the mean of the distribution,
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/k/ is the random variable, /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 = e[super -[lambda]] [lambda][super k] \/ k! ]]
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[[logpdf][log(pdf) = -lgamma(k+1) + k*log([lambda]) - [lambda] if k > 0 and [lambda] > 0 ]]
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[[cdf][Using the relation: p = [Gamma](k+1, [lambda]) \/ k! = __gamma_q(k+1, [lambda])]]
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[[cdf complement][Using the relation: q = __gamma_p(k+1, [lambda]) ]]
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[[quantile][Using the relation: k = __gamma_q_inva([lambda], p) - 1]]
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[[quantile from the complement][Using the relation: k = __gamma_p_inva([lambda], q) - 1]]
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[[mean][[lambda]]]
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[[mode][ floor ([lambda]) or [floorlr[lambda]] ]]
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[[skewness][1/[radic][lambda]]]
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[[kurtosis][3 + 1/[lambda]]]
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[[kurtosis excess][1/[lambda]]]
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]
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[endsect] [/section:poisson_dist Poisson]
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[/ poisson.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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