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HardProbability Random Variables Expected Value~8m

Draw $X_1, X_2, \dots$ independently from $\text{Uniform}(0,1)$ and let $N = \min\{\,n : X_1 + \dots + X_n > 1\,\}$ , with $S_N = X_1 + \dots + X_N$ the sum at that moment. The value $\mathbb{E}[S_N]$ can be written as $c\,e$ for a rational $c$ . Find $c$ .

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First recall $\mathbb{E}[N]=e$ (since $\mathbb{P}(N>n)=1/n!$ ).
Apply Wald's identity: $\mathbb{E}[S_N]=\mathbb{E}[N]\,\mathbb{E}[X_1]$ with $\mathbb{E}[X_1]=1/2$ .

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Asked at: Jane Street, Two Sigma

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