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5984概率简单数值题short

Expected Inspection Cost Until the First Defect

题目

A quality line inspects items one at a time; each item is defective independently with probability 0.050.05. Inspection stops at the first defective item. Each inspection (defective or not) costs an i.i.d. amount CiC_i with E[Ci]=$8E[C_i]=\$8, independent of the defect outcomes. Let NN be the number of items inspected. Find the expected total inspection cost E ⁣[i=1NCi]E\!\left[\sum_{i=1}^N C_i\right].

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