discrete random variable

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) in- creases with x as long as x np – (1 – p). Conclude that the mode x* is the integer satis- fying (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.