We have seen that if E(X1) E(X2) ... E(Xn) m, then E(X1 ... Xn) nm. In some applications, the number of Xi

’s under consideration is not a fixed number n but instead is an rv N. For example, let N the number of components that are brought into a repair shop on a particular day, and let Xi denote the repair shop time for the ith component. Then the total repair time is X1 X2 . . . XN, the sum of a random number of random variables. When N is independent of the Xi

’s, it can be shown that E(X1 ... XN) E(N)  m

a. If the expected number of components brought in on a particularly day is 10 and expected repair time for a randomly submitted component is 40 min, what is the expected total repair time for components submitted on any particular day?

b. Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour.

The expected number of defects per component is 3.5.

What is the expected value of the total number of defects on components submitted for repair during a 4-hour period? Be sure to indicate how your answer follows from the general result just given.