Let's denote a binomial distribution with n trials and p the chance of success as Binom(n, p). Let's say a function f is a probability mass function of Binom(n, p), that is, f(x) = n x p x (1 p) nx for x = 0, 1, . . . , n.

(a) [3pts] Solve the inequality f(x) f(x 1). That is, find a condition of x which is equivalent to f(x) f(x 1). Recall that x is an integer such that 0 x n. Note 1: If you put x = 0 into the given inequality, you will see the term f(1). However, you cannot put -1 into the pmf of the binomial distribution. Thus, let's just define f(1) = 0. Note 2 (Example of solving an inequality): If you solve the inequality (x 2)(x 4) 0, you have 2 x 4. (

b) [2pts] Using the result of (a), show that x maximized f(x) is x = (n + 1)p. The meaning of (n + 1)p is the largest integer which is less than or equal to (n + 1)p. For example, 2.3 = 2, 3.9 = 3, and 10 = 10.