# CompoundPoissonDistribution

CompoundPoissonDistribution[λ,dist]

represents a compound Poisson distribution with rate parameter λ and jump size distribution dist.

# Details

- CompoundPoissonDistribution is also known as a stopped‐sum distribution.
- The compound Poisson distribution is equivalent to , where are independent and identically distributed random variables following dist and nPoissonDistribution[λ].
- CompoundPoissonProcess allows λ to be any positive real number and dist to be any univariate distribution.
- CompoundPoissonDistribution can be used with such functions as Mean, Variance, and RandomVariate.

# Background & Context

- CompoundPoissonDistribution[λ,dist] represents a discrete statistical distribution parameterized by a positive real number λ and a univariate distribution dist, the latter of which can be either discrete or continuous. The compound Poisson distribution models the sum of independent and identically distributed random variables , where X
_{i}dist for all and NPoissonDistribution[λ]. The parameters λ and dist determine all properties possessed by the probability density function (PDF) of a compound Poisson distribution, including its shape, height, location, and domain. The compound Poisson distribution is referred to by a variety of other terms, including Poisson-stopped sum, generalized Poisson distribution, multiple Poisson distribution, composed Poisson distribution, stuttering Poisson distribution, clustered Poisson distribution, Pollaczek–Geiringer distribution, and Poisson power series distribution. - The study of the compound Poisson process (under the name of Pollaczek–Geiringer distributions) dates back to the 1930s. In its infancy, the compound Poisson distribution was devised as a tool to model the statistical behavior of "rare events," including accidents, diseases, and suicides. Within the study of stochastic processes, the compound Poisson distribution is also the motivation behind the so-called Bernoulli process, a continuous-time stochastic process with jumps whose sizes are randomly distributed according to a specified distribution and in which the jumps arrive according to a Poisson process. More recently, stopped sum distributions such as the compound Poisson distribution have been used to model a variety of phenomena, including the types/frequencies of insurance claims and the frequencies/amounts of rainfall.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a compound Poisson distribution. Distributed[x,CompoundPoissonDistribution[λ,dist]], written more concisely as xCompoundPoissonDistribution[λ,dist] , can be used to assert that a random variable x is distributed according to a compound Poisson 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[CompoundPoissonDistribution[λ,dist],x] and CDF[CompoundPoissonDistribution[λ,dist],x], though the PDF (as well as "PDF-related" quantities such as HazardFunction and Likelihood) will be undefined whenever dist is a continuous distribution. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively and are defined for either continuous or discrete dist.
- DistributionFitTest can be used to test if a given dataset is consistent with a compound Poisson distribution, EstimatedDistribution to estimate a compound Poisson parametric distribution from given data, and FindDistributionParameters to fit data to a compound Poisson distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic compound Poisson distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic compound Poisson distribution.
- TransformedDistribution can be used to represent a transformed compound Poisson 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 compound Poisson distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving compound Poisson distributions.
- CompoundPoissonDistribution is related to a number of other statistical distributions and constructs. CompoundPoissonDistribution is a generalization of PoissonDistribution, and because of the allowance of the parameter dist to take on any univariate distribution, there exists a generic relationship between CompoundPoissonDistribution and the collection of all univariate distributions in the Wolfram Language. In a less abstract sense, CompoundPoissonDistribution is a slice distribution of CompoundPoissonProcess in the sense that CompoundPoissonProcess[λ,dist][t] simplifies to CompoundPoissonDistribution[t λ,dist]. In addition, several distributions within the Wolfram Language can be derived via CompoundPoissonDistribution[λ,dist] for various values of dist, e.g. BinomialDistribution (when dist is BernoulliDistribution) and NegativeBinomialDistribution (when dist is LogSeriesDistribution).

Introduced in 2012

(9.0)