Compound Poisson distribution

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. In the simplest cases, 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 ${\displaystyle N}$ i.i.d. random variables conditioned on the number of these variables (${\displaystyle N}$):

has 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.

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