Compound Poisson distribution

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In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose

$N\sim\operatorname{Poisson}(\lambda),$

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and

$X_1, X_2, X_3, \dots$

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum

$Y=\sum_{n=1}^N X_n$

is a compound Poisson distribution. (When N = 0, then the value of Y is 0.)

In terms of the basic moments,

$E(Y) = E(X) E(N)\,$

var(Y) = E (N)var(X) + E (X) 2 var(N)

Via the law of total cumulance it can be shown that the moments of X₁ are the cumulants of Y.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.

[edit] Compound Poisson processes

A compound Poisson process with rate $λ > 0$ and jump size distribution G is a continuous-time stochastic process $\{\,Y(t) : t \geq 0 \,\}$ given by

$Y(t) = \sum_{i=1}^{N(t)} D_i$

where, $\{\,N(t) : t \geq 0\,\}$ is a Poisson process with rate $λ$ , and $\{\,D_i : i \geq 0\,\}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\{\,N(t) : t \geq 0\,\}.\,$

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Category: Probability theory

Compound Poisson distribution

From Wikipedia, the free encyclopedia

[edit] Compound Poisson processes

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