Statistica Sinica 3(1993), 295-312

ON THE NUMBER OF SUCCESSES

IN INDEPENDENT TRIALS

Y. H. Wang

Concordia University

Abstract: A unified combinatorial approach is used to obtain many theorems about S n, the number of successes in n independent non-identical Bernoulli trials. The following results are, in particular, proved: (1) The variance of S n 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.

Key words and phrases: Bernoulli trials, Poisson's binomial distribution, unimodality, Poisson' theorem, total variation.