# PoissonConsulDistribution

represents a PoissonConsul distribution with parameters μ and λ.

# Details • PoissonConsulDistribution is also known as generalized Poisson distribution (GPD).
• The probability for integer value in a PoissonConsul distribution is for , and is zero for .
• PoissonConsulDistribution allows μ to be any positive real number and λ to be any real number between 0 and 1.
• PoissonConsulDistribution allows μ and λ to be dimensionless quantities. »
• PoissonConsulDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• represents a discrete statistical distribution defined for integer values and determined by the real parameters μ ( ) and λ ( ). The Poisson-Consul distribution has a probability density function (PDF) that is discrete and unimodal. The distribution is sometimes also referred to as the generalized Poisson distribution or GPD due to the fact that "the" Poisson distribution (PoissonDistribution) may be realized as a special case, and is also sometimes referred to as the Lagrangian Poisson distribution.
• The Poisson-Consul distribution is a generalization of the standard Poisson distribution (named after French mathematician Siméon Poisson) and bears the name of statistician Prem C. Consul who worked on the generalized distribution throughout the 1970s and 1980s. Like the classical Poisson distribution, the Poisson-Consul distribution is extremely useful in modeling situations consisting of a large number of independent trials with very small probability of occurrence (e.g. the number of deaths per year of cavalry soldiers stemming from kicks by horse) and extends the classical Poisson distribution in the sense that the probability of occurrence of a single event need remain constant. The Poisson-Consul distribution has also been used to model a number of modern phenomena such as domestic violence, and is a valuable modeling tool in areas such as finance and actuarial science.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Poisson-Consul distribution. Distributed[x,PoissonConsulDistribution[μ,λ]], written more concisely as xPoissonConsulDistribution[μ,λ], can be used to assert that a random variable x is distributed according to a Poisson-Consul distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[PoissonConsulDistribution[μ,λ],x] and CDF[PoissonConsulDistribution[μ,λ],x], though one should note that there is no closed-form expression for its PDF. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
• DistributionFitTest can be used to test if a given dataset is consistent with a Poisson-Consul distribution, EstimatedDistribution to estimate a Poisson-Consul parametric distribution from given data, and FindDistributionParameters to fit data to a Poisson-Consul distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Poisson-Consul distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Poisson-Consul distribution.
• TransformedDistribution can be used to represent a transformed Poisson-Consul distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Poisson-Consul distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Poisson-Consul distributions.
• PoissonConsulDistribution is related to a number of other statistical distributions. It is an immediate generalization of PoissonDistribution in that the PDF of is precisely the same as that of . In addition, PoissonConsulDistribution can be realized as a parameter mixture (ParameterMixtureDistribution) of BorelTannerDistribution while limits to InverseGaussianDistribution as μ (provided that μ (1-λ) remains fixed). PoissonConsulDistribution is also related to PascalDistribution, BinomialDistribution, NegativeBinomialDistribution, MultinomialDistribution, and NegativeMultinomialDistribution.

# Examples

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## Basic Examples(3)

Probability mass function:

 In:= Out= In:= Out= In:= Out= Cumulative distribution function:

 In:= Out= In:= Out= Mean and variance:

 In:= Out= In:= Out= ## Properties & Relations(5)

Introduced in 2010
(8.0)
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Updated in 2016
(10.4)