# Question

Let U1, U2, . . . be a sequence of independent uniform (0, 1) random variables. In Example 5i we showed that, for 0 ≤ x ≤ 1,E[N(x)] = ex, where

This problem gives another approach to establishing that result.

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

P{N(x) ≥ n + 1} = xn/n!

First condition on U1 and then use the induction hypothesis.

(b) Use part (a) to conclude that

E[N(x)] = ex

This problem gives another approach to establishing that result.

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

P{N(x) ≥ n + 1} = xn/n!

First condition on U1 and then use the induction hypothesis.

(b) Use part (a) to conclude that

E[N(x)] = ex

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