probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

and statistics, the geometric Poisson distribution (also called the Pólya–Aeppli distribution) is used for describing objects that come in clusters, where the number of clusters follows a

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

and the number of objects within a cluster follows a

geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \; * ...

. It is a particular case of the

compound Poisson distribution In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. ...

. The probability mass function of a random variable ''N'' distributed according to the geometric Poisson distribution

\mathcal(\lambda,\theta)

is given by :

f_N(n) = \mathrm(N=n)= \begin \sum_^n e^\frac(1-\theta)^\theta^k\binom, & n>0 \\ e^, & n=0 \end

where ''λ'' is the parameter of the underlying

and θ is the parameter of the geometric distribution. The distribution was described by

George Pólya George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...

in 1930. Pólya credited his student

Alfred Aeppli Alfred Aeppli was a Swiss mathematician. The Pólya–Aeppli distribution in probability theory and statistics is named after him and his doctoral advisor George Pólya. Life and work Alfred Aeppli was born in Zürich on 15 July 1894 to Alfred ...

's 1924 dissertation as the original source. It was called the geometric Poisson distribution by Sherbrooke in 1968, who gave probability tables with a precision of four decimal places. The geometric Poisson distribution has been used to describe systems modelled by a

Markov model In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Mark ...

, such as biological processes or traffic accidents.

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See also

References

Bibliography

Further reading