7. Suppose that k of the balls numbered 1, 2,..., n, where nk, are randomly chosen. Let X denote the maximum numbered ball chosen. Also, let R denote the number of the n k unchosen balls that have higher numbers than all the chosen balls.

(a) What is the relationship between X and R?

(b) Express Ras the sum of nk suitably defined Bernoulli random variables.

(e) Use parts

(a) and

(b) to find E[X].

(Note that E[X] was obtained previously in Theoretical Exercise 28 of Chap-

ter 4.)