On the nature of the binomial distribution

Abstract

We examine how the binomial distribution B(n, p) arises as the distribution S-n = Sigma (n)(i=1) X-i of an arbitrary sequence of Bernoulli variables. It is shown that B(n, p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2, p) and B(3, p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which S-n follows B (n, p), given that Sn-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n, p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.