Draws to pass a log threshold

HardProbability Random Variables Expected Value~8m

Draw $X_1, X_2, \dots$ independently from $\text{Uniform}(0,1)$ and let $N = \min\{\,n : X_1 + \dots + X_n > \ln 2\,\}$ be the number of draws needed for the running sum to first exceed $\ln 2$ . Find $\mathbb{E}[N]$ .

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Use $\mathbb{E}[N]=\sum_{n\ge0}\mathbb{P}(N>n)$ , and $N>n$ means the first $n$ draws sum to $\le \ln 2$ .
That simplex probability is $x^n/n!$ with $x=\ln 2$ . Summing gives $e^{x}$ .

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Draws to pass a log threshold

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