The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probablex 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 µ, the mode is the largest integer less than m. If m is an integer, show that both µ - 1 and µ are modes.