Characterization of the Compound Poisson Distribution

Consider two non-negative integer-valued r.v.'s X,Y with X=>Y. Suppose that the conditional distribution of Y|X is binomial with parameters (n,p), n=0,1,2,...; 0 < p < 1 and p independent of n. It is known, and can be checked easily, that under the above assumption the distribution of Y is Poisson with parameter λp, λ>0 (Poisson(λp)) if and only if (iff) X is Poisson (λ). This model has been extensively used in the literature under different names in many practical situations.