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

<math>N\sim\operatorname{Poisson}(\lambda),</math>

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

<math>X_1, X_2, X_3, \dots</math>

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

<math>Y=\sum_{n=1}^N X_n</math>

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

In terms of the basic moments,

<math>E(Y) = E(X) E(N)\,</math>

<math>\operatorname{Var}(Y) = E(N)\operatorname{Var}(X) + {E(X)}^2 \operatorname{Var}(N)</math>

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

<math>\operatorname{Var}(Y) = E(N)(\operatorname{Var}(X) + {E(X)}^2 )</math>

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 <math>\lambda>0</math> and jump size distribution G is a continuous-time stochastic process <math>\{\,Y(t) : t \geq 0 \,\}</math> given by

<math>Y(t) = \sum_{i=1}^{N(t)} D_i</math>

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

it:Distribuzione composta di Poisson