81. Let Xi , i1, be independent uniform (0, 1) random variables, and define N by N = min{n: Xn < Xn−1}

where X0 = x. Let f (x) = E[N].

(a) Derive an integral equation for f (x) by conditioning on X1.

(b) Differentiate both sides of the equation derived in part (a).

(c) Solve the resulting equation obtained in part (b).

(d) For a second approach to determining f (x) argue that P{N  k} = (1 − x)k−1

(k − 1)!

(e) Use part

(d) to obtain f (x).