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Q56Section C (Section C)56 of 60

Expected value of random sum of Bernoullis

Let X1,X2,…,X5X_1, X_2, \dots, X_5 be i.i.d. random variables, each having the Bin(1,12)\mathrm{Bin}(1, \frac{1}{2}) distribution. Let K=X1+X2+⋯+X5K = X_1+X_2+\dots+X_5 and U={0,if K=0, X1+X2+⋯+XK,if K=1,2,…,5.U=\begin{cases} 0, & \text{if }K=0, \ X_1+X_2+\dots+X_K, & \text{if }K=1, 2, \dots, 5. \end{cases} Then, E(U)E(U) equals __________

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