Hard round for quantitative research.
Uniforms until the sum passes one
The question
You draw independent uniform numbers on [0,1] and stop when their sum first exceeds 1. How many draws do you expect to make?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
e, about 2.718.
The probability that the sum of n uniforms stays below 1 is 1/n!, the volume of the standard simplex.
The number of draws N exceeds n exactly when the first n sum to less than 1, so P(N > n) = 1/n!.
Expectation is the sum over n >= 0 of P(N > n), which is the sum of 1/n!, equal to e.
It is a rare closed form and interviewers like it because the derivation is short once you see that the simplex volume is the whole problem.
What it is testing
Tail-sum expectation plus recognising a simplex volume.
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