132. The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probable x value).

a. Let X  Bin(n, p). By considering the ratio b(x  1; n, p)/b(x; n, p), show that b(x; n, p) increases with x as long as x  np  (1  p).

Conclude that the mode x* is the integer satisfying

(n  1)p  1 x*

(n  1)p.

b. Show that if X has a Poisson distribution with parameter l, the mode is the largest integer less than l. If l is an integer, show that both l  1 and l are modes.