93. Consider a sequence of independent trials, each of which is equally likely to result in any of the outcomes 0, 1,..., m. 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 1,..., m −1 have occurred in the same round. Also, let Tj denote the number of trials that it takes until j distinct outcomes have occurred, and let Ij denote the jth distinct outcome to occur. (Therefore, outcome Ij first occurs at trial Tj .)

(a) Argue that the random vectors(I1,..., Im) and (T1,..., Tm) are independent.

(b) Define X by letting X = j if outcome 0 is the jth distinct outcome to occur. (Thus, IX = 0.) Derive an equation for E[N] in terms of E[Tj], j =

1,..., m − 1 by conditioning on X.

(c) Determine E[Tj], j = 1,..., m − 1.

Hint: See Exercise 42 of Chapter 2.

(d) Find E[N].