CompoundPoissonDistribution

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`CompoundPoissonDistribution`

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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 nPoissonDistribution[λ].
CompoundPoissonDistribution 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 NPoissonDistribution[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 xCompoundPoissonDistribution[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).

Examples

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Basic Examples (2)Summary of the most common use cases

Define a compound Poisson distribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-e0psju

Mean and variance:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-j8era2

Out[2]=2

Simulate a compound Poisson distribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ed8qmg

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ufc50

Out[2]=2

Scope (7)Survey of the scope of standard use cases

Simulate a compound Poisson distribution for different renewal rates:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ngvjxq

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-wu72es

Out[2]=2

Distribution parameter estimation:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-3mgdcw

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-xhx5e3

Out[2]=2

Compare the histogram of the data with a sample from the estimated distribution:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-nt8cl0

Out[3]=3

Define a compound Poisson distribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-jqvbz8

Skewness:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-f7rb7p

Out[2]=2

Kurtosis:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-tlm11p

Out[3]=3

Define a compound Poisson distribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-gncucz

Calculate cumulants:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-hl9wpo

Out[2]=2

Compare to the reward distribution:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-9odc59

Out[3]=3

Compare to the renewal distribution:

In[4]:=4

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-1365se

Out[4]=4

Different moments with closed forms as functions of parameters:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-osmyj7

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-js043h

Moment:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-rx074o

Out[3]=3

CentralMoment:

In[4]:=4

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-pknsqa

Out[4]=4

FactorialMoment:

In[5]:=5

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-zg9ct4

Out[5]=5

Cumulant:

In[6]:=6

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-9gzmth

Out[6]=6

Generating functions for a CompoundPoissonDistribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-fybfnv

Characteristic function:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-lp5wcf

Out[2]=2

Moment generating function:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-oaxbcq

Out[3]=3

Factorial moment generating function:

In[4]:=4

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-mbt0cc

Out[4]=4

Cumulant generating function:

In[5]:=5

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-j42h09

Out[5]=5

Compound Poisson distribution with a mixture jump distribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-e6y6zt

Mean of the distribution:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ch4bl0

Out[2]=2

Compare with the value obtained from simulation:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-i7bj4p

Out[3]=3

Applications (2)Sample problems that can be solved with this function

Shoppers arrive at a newly renovated store according to a Poisson process with a rate of 20 customers per hour. The store promotes this event by giving every customer a gift. The gift has a value that follows a Weibull distribution with shape parameter 10 and scale parameter 3. Find the expected total cost of the gifts given by the store during the 12-hour period for which the store is open on that day:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-h79xr3

Expected total cost of the gifts given on the inaugural day:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-bbqchm

Out[2]=2

Probability that the store spends between $500 and $800 on the gifts:

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-bc2s7l

In[4]:=4

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-cgdtmb

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-cwbsqu

Out[5]=5

Aggregate claims from a risk have a compound Poisson distribution with Poisson parameter 200 and an individual claim amount distribution, which is a Pareto distribution with minimum value parameter 300, shape parameter 3, and location parameter 0. The insurer has effected excess of loss reinsurance with retention level $300. Calculate the mean and variance of the reinsurer's aggregate claims. The individual claims for the reinsurer follow a transformed Pareto distribution, since the reinsurer only pays amounts greater than $300:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-guvasg

The aggregate claims:

In[2]:=2

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ee872y

In[3]:=3

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-cjfyv3

Out[3]=3

Mean and variance of the reinsurer's aggregate claims:

In[4]:=4

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-ba96ln

Out[4]=4

Properties & Relations (5)Properties of the function, and connections to other functions

CompoundPoissonDistribution is a slice distribution of CompoundPoissonProcess:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-e79vr5

Out[1]=1

CompoundPoissonDistribution with BernoulliDistribution is a PoissonDistribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-4qoyzp

Out[1]=1

CompoundPoissonDistribution with a special BorelTannerDistribution follows PoissonConsulDistribution:

In[1]:=1

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https://wolfram.com/xid/0gcqvd9kgppytchrbu-lr1t0z

The sum of Borel–Tanner distributed variables follows the Borel–Tanner distribution, hence the slice distribution is equivalent to the parameter mixture distribution:

In[2]:=2