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452.Uniform Sum Past Two

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expected valueCh. 6 — Random Variables and Expectation444+ Problems in Probability
Let X1,X2,⋯∼Unif(0,1)X_1, X_2, \dots \sim \mathrm{Unif}(0,1) IID and let N2=min⁡{n:X1+⋯+Xn>2}N_2 = \min\{n : X_1 + \cdots + X_n > 2\} be the first time the running sum exceeds 2. Find E[N2]E[N_2].

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