The negative binomial random variable Y was defined in Section 3.6 as the number of the

Chapter 5, Problem 5.159

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The negative binomial random variable Y was defined in Section 3.6 as the number of the trial on which the rth success occurs, in a sequence of independent trials with constant probability p of success on each trial. Let Xi denote a random variable defined as the number of the trial on which the ith success occurs, for i = 1, 2,...,r. Now define Wi = Xi Xi1, i = 1, 2,...,r, where X0 is defined to be zero. Then we can write Y = r i=1 Wi . Notice that the random variables W1, W2,..., Wr have identical geometric distributions and are mutually independent. Use Theorem 5.12 to show that E(Y ) = r/p and V(Y ) = r(1 p)/p2.