Let U1 U2 be a sequence of
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
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