We have seen that if E(X) = E(X) == E(X) = , then E(X++ X) = nu. In some applications, the number of X's under consideration is not a fixed num- bern 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 X, denote the repair shop time for the ith component. Then the total repair time is X +X++Xy, the sum of a random num- ber of random variables. When N is independent of the X's, it can be shown that E(X++Xx) = E(N) p

a. If the expected number of components brought in on a particularly day is 10 and expected repair time for a ran- domly 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.