5.23 A random variable X has the negative binomial distribution, pr(X = k) = n + k − 1 n − 1

 p q

k 

1 − p q

n

, for k ≥ 0, in which q = 1+ p. Show that the expectation and variance of X are given by E[X] = np, Var(X) = np(1 + p). Use these expressions to derive estimates of n and p using the method of moments.