Statistica Sinica

Y. H. Wang

Abstract:A unified combinatorial approach is used to obtain many theorems aboutS_{n}, the number of successes innindependent non-identical Bernoulli trials. The following results are, in particular, proved: (1) The variance ofS_{ n}increases as the set of success probabilities {p_{i}} tends to be more and more homogeneous and attains its maximum as they become identical; (2) The density ofS_{n}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 ofS_{n}and that of the Poisson.

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