Number of uniforms to sum>1

[Sorry - been a long time without posting. Illness has made it hard to find the impetus.]


Anyway, a probability problem I found interesting:

What's the distribution of the number of random uniform[0,1) random variables that you need to generate so that the sum is greater than 1?

There's a nifty little trick to it.

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Update:

Well, the trick is to compute the survivor function, 1-F(n).

Let N be the number of uniform RVs needed to exceed 1.

P(N>n) = P(U(1) + U(2) + ... + U(n) <= 1)

That RHS probability is relatively easy to get in a variety of ways to be 1/n!

From there, the probability function for N is simple...