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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.

Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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