Characterization of the inflated Poisson distribution

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Suppose that the random number X of particles entering a system has the distribution pn=P{X=n}, n=0,1,⋯. Due to the interference of noise, some of the arriving particles are not registered, and the number of particles observed is actually Y, where Y≤X, and Y has the distribution qn=P{Y=n}, n=0,1,⋯. The interference of noise is characterized by the conditional probabilities s(r,n)=P{Y=r|X=n}, 0≤r≤n. In this paper the author assumes that the distribution of X belongs to the class PSD of power series distributions and that the relations E(Y|X=n)=αnp and E(Y2|X=n)=αnp(1−p)+αn2p2 hold with 0