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 A, (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 ran- domly selected customer is Ax; A, AP) = {P,e^x + (1 otherwise This is often called the hyperexponential or mixed expo- nential distribution. This distribution is also proposed as a model for rainfall amount in "Modeling Monsoon Affected Rainfall of Pakistan by Point Processes" (J. of Water Re- sources Planning and Mgmnt., 1992: 671-688).

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

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

c. If X has fix; A, A, p) as its pdf, what is E(X)?

d. Using the fact that E(X) =2/A2 when X has an expo- nential distribution with parameter A, compute E(X) when X has pdf f(x; A1, A2, p). Then compute V(X).

e. The coefficient of variation of a random variable (or distribution) is CV =/u. What is CV for an exponen- tial 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 A 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.]