How many uniforms until the sum passes one

Question

Keep drawing independent uniform random numbers on $[0,1]$ and adding them up, stopping as soon as the running sum is greater than $1$ . On average, how many numbers do you draw?

Short answer

$e\approx2.718$ . The first $n$ draws sum to at most $1$ with probability $\tfrac{1}{n!}$ (volume of a simplex), so $P(N>n)=\tfrac1{n!}$ and $E[N]=\sum_{n\ge0}\tfrac1{n!}=e$ .

Adapted from Wilmott, Frequently Asked Questions in Quantitative Finance, ch. 10 brainteaser 'Sums of uniform random variables', p.351; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Independence
Conditional Expectation and Variance Revisited

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