# 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

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

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

${\displaystyle X_{1},X_{2},X_{3},\dots }$

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

${\displaystyle Y=\sum _{n=1}^{N}X_{n}}$

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

In terms of the basic moments,

${\displaystyle E(Y)=E(X)E(N)\,}$
${\displaystyle \operatorname {Var} (Y)=E(N)\operatorname {Var} (X)+{E(X)}^{2}\operatorname {Var} (N)}$

as E(N)=Var(N) if N is Poisson it can be reduced to

${\displaystyle \operatorname {Var} (Y)=E(N)(\operatorname {Var} (X)+{E(X)}^{2})}$

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.

## Compound Poisson processes

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

${\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}$

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