Expected Number of Uniforms to Exceed One
Let $U_1, U_2, U_3, \ldots$ be i.i.d. $\text{Uniform}(0,1)$ random variables. Define the stopping time:
$N = \min\{n : U_1 + U_2 + \cdots + U_n > 1\}$
What is $E[N]$?
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