The full distribution of draws to exceed one

Question

Draw U1, U2, \dots independently from U(0,1) and let N be the smallest n with U1 + \cdots + Un 1. It is known that E[N] = e. Find the exact distribution P(N = n) and the variance \operatorname{Var}(N).

Accepted Answer

The distribution. With P(N n) = P(U1 + \cdots + Un \le 1) = 	frac{1}{n!}, P(N = n) = P(N n-1) - P(N n) = \frac{1}{(n-1)!} - \frac{1}{n!} = \frac{n-1}{n!}, \qquad n \ge 2. Since a single draw is always \le 1, N \ge 2 always; indeed P(N=1) = 0/1 = 0. Check the smallest cases: P(N=2) = 	frac{1}{2}, P(N=3) = 	frac{2}{6} = 	frac13, P(N=4) = 	frac{3}{24} = 	frac18. These sum (with the rest) to 1, and \sumn n\,P(N=n) = e recovers the mean. The variance. Compute the factorial moment E[N(N-1)]: E[N(N-1)] = \sum{n\ge 2} n(n-1)\frac{n-1}{n!} = \sum{n\ge 2}\frac{n-1}{(n-2)!}. Substituting m = n-2 g…

The full distribution of draws to exceed one

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