Interrelations among Generalized Distributions and Their Components

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Abstract
If X1 and X2 are discrete random variables with probability generating functions (p. g. f.) gl{g2(z)} is called X1 - generalized - X2 and denoted by X1 v X2. Fractorial moments of X1 v X2 are expressed in terms of the factorial moments of the components X1 and X2, and the factorial cumulants in terms of the factorial cumulants of X1 and factorial moments of X2. Twenty-two special distributions were derived by substituting specific functions for gj (z) and g2 (z) and it was discovered that a distribution with p. g. f. 1-p1 log {q2-p2exp [ (z-1)]}; q2=1+q2; called the Log-Zero-Poisson has more flexibility than all the other special distributions.

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