Expected total once the uniforms first pass one

Question

Draw U1, U2, \dots independently from U(0,1) and let N be the smallest n with SN = U1 + \cdots + Un 1. What is E[SN], the expected value of the running sum at the moment it first exceeds 1? What is the expected overshoot beyond 1?

Accepted Answer

Because you stop the instant the sum crosses 1, the decision to stop at step n depends only on U1, \dots, Un, so N is a stopping time with finite mean E[N] = e. Wald's identity then applies: for i.i.d. draws with mean \mu and a stopping time N, E[SN] = E[N]\,\mu. Here \mu = E[U] = 	frac12 and E[N] = e, so E[SN] = e \cdot \frac12 = \frac{e}{2} \approx 1.359. The expected overshoot past the threshold is therefore E[SN - 1] = \frac{e}{2} - 1 \approx 0.359. A sanity check: the sum must exceed 1, so E[SN] 1, and the last draw added is on average a bit above half (draws that push you over tend to …

Expected total once the uniforms first pass one

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