Characterization of Discrete Probability Distributions by Partial Independence.

If X and Y are random variables such that P X Y 1 and the conditional distribution of Y given X is binomial, then Moran 1952 showed that Y and X-Y are independent if X is Poisson. This document extends Morans result to a more general type of conditional distribution of Y given X, using only partial independence of Y and X-Y. This provides a generalization of a recent results of Janardhan and Rao 1982 on the characterization of generalized Polya-Eggenberger distribution. A variant of Morans theorem is proved which generalizes the results of Patil and Seshadri 1964 on the characterization of the distribution of a random variable x based on some conditions on the conditional distribution of Y given X and the independence of Y and X-Y.