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/PIX k-1] and see for what values of k it is greater or less than 1.