68. The special case of the gamma distribution in which  is a positive integer n is called an Erlang distribution. If we replace
by 1/ in Expression (4.8), the Erlang pdf is f(x; , n)





(

(



n x



)n

1 1

e

)



!

x

 x  0 0 x  0 It can be shown that if the times between successive events are independent, each with an exponential distribution with parameter , then the total time X that elapses before all of the next n events occur has pdf f(x; , n).

a. What is the expected value of X? If the time (in minutes)

between arrivals of successive customers is exponentially distributed with 
.5, how much time can be expected to elapse before the tenth customer arrives?

b. If customer interarrival time is exponentially distributed with 
.5, what is the probability that the tenth customer (after the one who has just arrived) will arrive within the next 30 min?

c. The event {X  t} occurs iff at least n events occur in the next t units of time. Use the fact that the number of events occurring in an interval of length t has a Poisson distribution with parameter t to write an expression (involving Poisson probabilities) for the Erlang cdf F(t; , n)

P(X  t).