16. Let X have the negative binomial distribution with parameters r and p. Thus, X counts the total number of Bernoulli trials until the rth success. Next, let X1 denote the number of trials up to and including the first success, let X2 denote the number from the first success up to and including the second success, and so on, so that Xr denotes the number of trials from the (r − 1)-st success up to and including the rth success. Note that the Xi’s are iid having the geometric distribution, X = X1 + ·· ·+Xr.

Use Corollary 4.4-1 and Proposition 4.4-4 to derive the expected value and variance of the negative binomial random variable X. (Hint. The expected value and variance of a geometric random variable are derived in Examples 3.3-3 and 3.3-12.)