20. 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 k −1 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 k −1 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.)