15. Let X be binomially distributed with parameters n and p. Show that as k goes from 0 to n, P(X = k) increases monotonically, then decreases monotonically reaching its largest value

(a) in the case that (n + 1)p is an integer, when k equals either (n + 1)p − 1 or

(n + 1)p,

(b) in the case that (n + 1)p is not an integer, when k satisfies (n + 1)p−1 < k <

(n + 1)p.

Hint: Consider P{X = k}/P{X = k − 1} and see for what values of k it is greater or less than 1.