Recently the authors, Cacoullos and Papageorgiou (1982) and Papageorgiou (1982) , have been doing some work on characterizing a discrete random variable Y which has a mixture representation in terms of another discrete random variable X(t), where the parameter t plays the role of the mixing variable. Of special interest is the case of a Poisson process {X(t), t > O} where. the time parameter t may be regarded as a random variable. The probability function p (y) of Y, in such a case, is given by the Poisson mixture