Independent trials, each of which is a success with

Chapter , Problem 20

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Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k1 trials are all failures and the kth a success. X is called a geometric random variable. Compute (a) P{X = k}, k = 1, 2, . . . ; (b) E[X]. Let Y denote the number of trials needed to obtain r successes. Y is called a negative binomial random variable. Compute (c) P{Y = k}, k = r, r + 1, . . . . (Hint: In order for Y to equal k, how many successes must result in the first k1 trials and what must be the outcome of trial k?) (d) Show that E[Y] = r/p (Hint: Write Y = Y1 + + Yr where Yi is the number of trials needed to go from a total of i 1 to a total of i successes.)