Compound Poisson distribution

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

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

i.e., N is a random variable is 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

$Y=\sum _{n=0}^{N}X_{n}$

is a compound Poisson distribution.

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