Characterization of the Compound Poisson Distribution

Abstract

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 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