BUILT-IN MATHEMATICA SYMBOL

PolyaAeppliDistribution

represents a Polya-Aeppli distribution with shape parameters

and p.

MORE INFORMATION

The Polya-Aeppli distribution is a compound geometric Poisson distribution, i.e. the distribution of a sum of independent identically distributed geometric random variates where the number of variates follows Poisson distribution.

The probability for positive integer value in a Polya-Aeppli distribution is proportional to .

PolyaAeppliDistribution allows to be any positive real number, and p is a number between 0 and 1.

PolyaAeppliDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (3)

Probability density function:

Cumulative distribution function:

Mean and variance:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Scope (7)

Generate a set of pseudorandom numbers that are Polya-Aeppli distributed:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness:

Limiting values:

Kurtosis:

Limiting values:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Closed form for symbolic order:

Cumulant:

Hazard function:

Quantile function:

Applications (2)

The CDF of PolyaAeppliDistribution is an example of a right-continuous function:

The number of hotbeds of a contagious disease follows PoissonDistribution with mean 10, while the number of sick people within the hotbed follows GeometricDistribution with mean 7. Find the probability that the total number of sick people is greater than 70:

Plot the distribution mass function for the number of sick people:

Properties & Relations (3)

Polya-Aeppli distribution is closed under addition:

Proof using characteristic functions:

Relationships to other distributions:

PoissonDistribution is a limiting case for Polya-Aeppli distribution: