Special Cases
When are non-negative discrete random variable,then this compound Poisson distribution is named stuttering Poisson distribution (non-negative discrete compound Poisson distribution) which can model batch arrivals (such as in a bulk queue).
If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the "compound Poisson distribution" outlined above is essentially the same as the more general class of compound probability distributions. However, the properties outlined above do depend on its formulation as the sum of a Poisson-distributed number of random variables. The distribution of Y in the case of the compound Poisson distribution with exponentially-distributed summands can be written in an form.
Read more about this topic: Compound Poisson Distribution
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