Bayesian Statistics: An Introduction, 4th Edition

A.11 Binomial distribution

X has a binomial distribution of index n and parameter π, denoted

if it has a discrete distribution with density

The mean and variance are

Because

we see that p(X+1)> p(X) if and only if

and hence that a mode occurs at

the square brackets denoting ‘integer part of’, and this mode is unique unless is an integer.

Integration by parts shows that the distribution function is expressible in terms of the incomplete beta function, namely,

see, for example, Kendall, Stewart and Ord (1987, Section 5.7).