15. Let X be binomially distributed with parameters and p. Show that as k goes from...

15. Let X be binomially distributed with parameters ç 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 + \)p is an integer, when k equals either

(n + l)p - 1 or (n + 1)/?,

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

(n + l)p - 1 < k < (n + l)p.

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