58. Let *U*₁, *U*₂,... be a sequence of independent uniform (0, 1) random variables.

In Example 4h we showed that for 0 ≤ *x* ≤ 1, *E[N(x)]* = *e^x*, where

$$N(x) = min{n: \sum_{i=1}^{n}U_{i} > x}$$

This problem gives another approach to establishing this result.

(a) Show by induction on *n* that for 0 < *x* ≤ 1 and all *n* ≥ 0,

$$P(N(x) ≤ n + 1) = \frac{x^{n}}{n!}$$

HINT: First condition on *U*₁ and then use the induction hypothesis.

(b) Use part

(a) to conclude that

$$E[N(x)] = e^{x}$$