*93. Consider a sequence of independent trials, each of which is equally likely to result in any of the outcomesI mage. Say that a round begins with the first trial, and that a new round begins each time outcome 0 occurs. Let N denote the number of trials that it takes until all of the outcomes have occurred in the same round. Also, letI mage denote the number of trials that it takes until j distinct outcomes have occurred, and let denote the jth distinct outcome to occur. (Therefore, outcome first occurs at trialI mage.)

(a) Argue that the random vectors and are independent.

(b) Define X by letting Image if outcome 0 is the jth distinct outcome to occur. (Thus, Image.) Derive an equation for in terms of by conditioning on X.

(c) Determine .

Hint: See Exercise 42 of Chapter 2.

(d) Find .