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).