Chapter 7: Functions of random variables

When developing a voice network, one measure of the availability of service is congestion. Congestion is the probability that the next call will be blocked. If there are c circuits, then congestion is the probability that c or more calls are in progress. A large sales firm has calls arriving according to a Poisson process with an average of 5 calls during any minute of the business day. From experience, they know that they will have congestion if 8 or more calls arrive during a minute. Let X be 1 if there is congestion during a randomly selected minute of the business day and 0 otherwise. a. Find the probability function of X. b. The firm believes that it loses $45 worth of customer goodwill during each minute of congestion. Define C, the total loss of customer goodwill during a randomly selected hour during the business day, in terms of X. Use this expression as well as the properties of expectation to solve E(C) and Var(C) c. Find the probability function of C.