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1 | [section:poisson_dist Poisson Distribution] |
| 2 | ||
| 3 | ``#include <boost/math/distributions/poisson.hpp>`` | |
| 4 | ||
| 5 | namespace boost { namespace math { | |
| 6 | ||
| 7 | template <class RealType = double, | |
| 8 | class ``__Policy`` = ``__policy_class`` > | |
| 9 | class poisson_distribution; | |
| 10 | ||
| 11 | typedef poisson_distribution<> poisson; | |
| 12 | ||
| 13 | template <class RealType, class ``__Policy``> | |
| 14 | class poisson_distribution | |
| 15 | { | |
| 16 | public: | |
| 17 | typedef RealType value_type; | |
| 18 | typedef Policy policy_type; | |
| 19 | ||
| 20 | poisson_distribution(RealType mean = 1); // Constructor. | |
| 21 | RealType mean()const; // Accessor. | |
| 22 | } | |
| 23 | ||
| 24 | }} // namespaces boost::math | |
| 25 | ||
| 26 | The [@http://en.wikipedia.org/wiki/Poisson_distribution Poisson distribution] | |
| 27 | is a well-known statistical discrete distribution. | |
| 28 | It expresses the probability of a number of events | |
| 29 | (or failures, arrivals, occurrences ...) | |
| 30 | occurring in a fixed period of time, | |
| 31 | provided these events occur with a known mean rate [lambda][space] | |
| 32 | (events/time), and are independent of the time since the last event. | |
| 33 | ||
| 34 | The distribution was discovered by Sim__eacute on-Denis Poisson (1781 to 1840). | |
| 35 | ||
| 36 | It has the Probability Mass Function: | |
| 37 | ||
| 38 | [equation poisson_ref1] | |
| 39 | ||
| 40 | for k events, with an expected number of events [lambda]. | |
| 41 | ||
| 42 | The following graph illustrates how the PDF varies with the parameter [lambda]: | |
| 43 | ||
| 44 | [graph poisson_pdf_1] | |
| 45 | ||
| 46 | [discrete_quantile_warning Poisson] | |
| 47 | ||
| 48 | [h4 Member Functions] | |
| 49 | ||
| 50 | poisson_distribution(RealType mean = 1); | |
| 51 | ||
| 52 | Constructs a poisson distribution with mean /mean/. | |
| 53 | ||
| 54 | RealType mean()const; | |
| 55 | ||
| 56 | Returns the /mean/ of this distribution. | |
| 57 | ||
| 58 | [h4 Non-member Accessors] | |
| 59 | ||
| 60 | All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all | |
| 61 | distributions are supported: __usual_accessors. | |
| 62 | ||
| 63 | The domain of the random variable is \[0, [infin]\]. | |
| 64 | ||
| 65 | [h4 Accuracy] | |
| 66 | ||
| 67 | The Poisson distribution is implemented in terms of the | |
| 68 | incomplete gamma functions __gamma_p and __gamma_q | |
| 69 | and as such should have low error rates: but refer to the documentation | |
| 70 | of those functions for more information. | |
| 71 | The quantile and its complement use the inverse gamma functions | |
| 72 | and are therefore probably slightly less accurate: this is because the | |
| 73 | inverse gamma functions are implemented using an iterative method with a | |
| 74 | lower tolerance to avoid excessive computation. | |
| 75 | ||
| 76 | [h4 Implementation] | |
| 77 | ||
| 78 | In the following table [lambda][space] is the mean of the distribution, | |
| 79 | /k/ is the random variable, /p/ is the probability and /q = 1-p/. | |
| 80 | ||
| 81 | [table | |
| 82 | [[Function][Implementation Notes]] | |
| 83 | [[pdf][Using the relation: pdf = e[super -[lambda]] [lambda][super k] \/ k! ]] | |
| 84 | [[cdf][Using the relation: p = [Gamma](k+1, [lambda]) \/ k! = __gamma_q(k+1, [lambda])]] | |
| 85 | [[cdf complement][Using the relation: q = __gamma_p(k+1, [lambda]) ]] | |
| 86 | [[quantile][Using the relation: k = __gamma_q_inva([lambda], p) - 1]] | |
| 87 | [[quantile from the complement][Using the relation: k = __gamma_p_inva([lambda], q) - 1]] | |
| 88 | [[mean][[lambda]]] | |
| 89 | [[mode][ floor ([lambda]) or [floorlr[lambda]] ]] | |
| 90 | [[skewness][1/[radic][lambda]]] | |
| 91 | [[kurtosis][3 + 1/[lambda]]] | |
| 92 | [[kurtosis excess][1/[lambda]]] | |
| 93 | ] | |
| 94 | ||
| 95 | [/ poisson.qbk | |
| 96 | Copyright 2006 John Maddock and Paul A. Bristow. | |
| 97 | Distributed under the Boost Software License, Version 1.0. | |
| 98 | (See accompanying file LICENSE_1_0.txt or copy at | |
| 99 | http://www.boost.org/LICENSE_1_0.txt). | |
| 100 | ] | |
| 101 | ||
| 102 | [endsect][/section:poisson_dist Poisson] | |
| 103 |