142. 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 li (i  1, 2) and p is the proportion of all customers served by facility 1, f 1x2  1.1 2e.20x0 q  x  q then the pdf of X  the service time of a randomly selected customer is 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 Manag., 1992: 671– 688).

a. Verify that f (x; l1, l2, p) is indeed a pdf.

b. What is the cdf F(x; l1, l2, p)?

c. If X has f (x; l1, l2, p) as its pdf, what is E(X)?

d. Using the fact that E(X2)  2/l2 when X has an exponential distribution with parameter l, compute E(X2) when X has pdf f (x; l1, l2, p).

Then compute V(X).

e. The coefficient of variation of a random variable

(or distribution) is CV  s/m. 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 l and n as defined in Exercise 76?

(Note: In applied work, the sample CV is used to decide which of the three distributions might be appropriate.)