math lessons - Compound Poisson distribution

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.)

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

01-04-2007 01:18:14
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