Solved: In some systems, a customer is allocated to one of

Chapter 4, Problem 4.121

(choose chapter or problem)

In some systems, a customer is allocated to one of two service facilities. If the service time for a customer served by facility i has an exponential distribution with parameter i (i 1, 2) and p is the proportion of all customers served by facility 1, then the pdf of X the service time of a randomly selected customer is f(x; 1, 2, p) p1e1 x (1 p)2e2 x x 0 0 otherwise This is often called the hyperexponential or mixed exponential distribution. This distribution is also proposed as a model for rainfall amount in Modeling Monsoon Affected Rainfall of Pakistan by Point Processes (J. Water Resources Planning and Mgmnt., 1992: 671688). a. Verify that f(x; 1, 2, p) is indeed a pdf. b. What is the cdf F(x; 1, 2, p)? c. If X has f(x; 1, 2, p) as its pdf, what is E(X)? d. Using the fact that E(X2 ) 2/2 when X has an exponential distribution with parameter , compute E(X2 ) when X has pdf f(x; 1, 2, p). Then compute V(X). e. The coefficient of variation of a random variable (or distribution) is CV /. What is CV for an exponential rv? What can you say about the value of CV when X has a hyperexponential distribution? f. What is CV for an Erlang distribution with parameters and n as defined in Exercise 68? [Note: In applied work, the sample CV is used to decide which of the three distributions might be appropriate.] 1