If Y is a negative binomial random variable, define Y = Y r. If Y is interpreted as the number of the trial on which the rth success occurs, then Y can be interpreted as the number of failures before the rth success. a If Y = Y r, P(Y = y) = P(Y r = y) = P(Y = y + r) for y = 0, 1, 2,..., show that P(Y = y) = y + r 1 r 1 pr qy , y = 0, 1, 2,.... b Derive the mean and variance of the random variable Y by using the relationship Y = Y r, where Y is negative binomial and the result in Exercise 3.33. 3.7 The Hypergeometric Probability Distribution 125 *3.101 a We observe a sequence of independent identical trials with two possible outcomes on each trial, S and F, and with P(S) = p. The number of the trial on which we observe the fifth success, Y , has a negative binomial distribution with parameters r = 5 and p. Suppose that we observe the fifth success on the eleventh trial. Find