Definition
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 conditioned on the number of these variables :
is a well-defined distribution. In the case N = 0, then the value of Y is 0, so that then Y | N=0 has a degenerate distribution.
The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, where this joint distribution is obtained by combining the conditional distribution Y | N with the marginal distribution of N.
Read more about this topic: Compound Poisson Distribution
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