On the Distribution of the Number of Successes in Independent Trials

Abstract

A unified combinatorial approach is used to obtain many theorems about Sn, the number of successes in n independent non-identical Bernoulli trials. The following results are, in particular, proved: (1) The variance of Sn increases as the set of success probabilities {pi} tends to be more and more homogeneous and attains its maximum as they become identical; (2) The density of Sn is unimodal: first increasing then decreasing; (3) Four different versions of Poisson's theorem; (4) An upper bound for the total variation between the distribution of Sn and that of the Poisson.