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Concept
Compound Poisson distribution
Summary
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution.
Suppose that
i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that
are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of i.i.d. random variables
is a compound Poisson distribution.
In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.
The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, and this joint distribution can be obtained by combining the conditional distribution Y | N with the marginal distribution of N.
The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus
Then, since E(N) = Var(N) if N is Poisson-distributed, 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, we have
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. And compound Poisson distributions is infinitely divisible by the definition.
When are positive integer-valued i.i.d random variables with , then this compound Poisson distribution is named discrete compound Poisson distribution (or stuttering-Poisson distribution) .
Official source
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