Chapter 3, Problem 116SE

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).

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