## Answers

Let the total number of boxes be T. Then T = 1 + X_2 + X_3 + ...+ X_12 Each X_i follows geometric distribution with theta = 1/12. Therefore E[X_i] = 1/theta = 12 var[X_i] = (1 - theta)/theta^2 = 132 mu_T = E[1 + X_2 + X_3 + ...+ X_12] = 1 + 12 + 12 + ... + 12 (11 times) = 133 var_T = var[1 + X_2 +X_3 + ...

.
+ X_12] = 0 + 11^2 times 132 Therefore sigma_T = squareroot 11^2 times 132 = 126.38 Let cost to complete be C. Then, C = 7.957 Therefore mu_c = 7.95 times mu_T = $ 1057.35 sigma_c = 7.95 times sigma_T = $ 1004.72

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