70. We have seen that if E(X1)E(X2). . .E(Xn)

m, then E(X1. . .Xn)nm. In some applications, g1Xi  Yi  Zi 2 the number of Xi s under consideration is not a xed number n but instead is an rv N. For example, let Nthe 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

a. If the expected number of components brought in on a particular 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.