8.15. Two independently operating launching procedures are used every week for launching rockets. Assume that each procedure is continued until it produces a successful launching. Suppose that using procedure I, P(S), the probability of a successful launching, equals p1, while for procedure II, P(S) = p2. Assume furthermore, that one attempt is made every week with each of the two methods. Let X1 and X2 represent the number of weeks required to achieve a successful launching by means of I and II, respectively. (Hence X1 and X2 are independent random variables, each having a geometric distri- bution.) Let W be the minimum (X1, X2) and Z be the maximum (X1, X2). Thus W represents the number of weeks required to obtain a successful launching while Z repre- sents the number of weeks needed to achieve successful launchings with both procedures. (Thus if procedure I results in 555S, while procedure II results in 35 S, we have W = 3, Z = 4.)

(a) Obtain an expression for the probability distribution of W. [Hint: Express, in terms of X1 and X2, the event (W=k}.]

(b) Obtain an expression for the probability distribution of Z.

(c) Rewrite the above expressions if p = p2.