Compound Poisson Distribution - Properties

Properties

Mean and variance of the compound distribution derive in a simple way from law of total expectation and the law of total variance. Thus

\operatorname{Var}_Y(Y) = E_N\left + \operatorname{Var}_N\left =\operatorname{E}_N\left + \operatorname{Var}_N\left ,

giving

\operatorname{Var}_Y(Y) = \operatorname{E}_N(N)\operatorname{Var}_X(X) + \left(\operatorname{E}_X(X)\right)^2\operatorname{Var}_N(N) .

Then, since E(N)=Var(N) if N is Poisson, and dropping the unnecessary subscripts, these formulae can be reduced to

The probability distribution of Y can be determined in terms of characteristic functions:

and hence, using the probability generating function of the Poisson distribution,

An alternative approach is via cumulant generating functions:

Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ=1, the cumulants of Y are the same as the moments of X1.

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

Read more about this topic:  Compound Poisson Distribution

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