The special case of the gamma distribution in which a isa positive integer n is called

Chapter 4, Problem 68

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

\(f(x ; \lambda, n)=\left\{\begin{array}{cc}\frac{\lambda(\lambda x)^{n-1} e^{-\lambda x}}{(n-1) !} & x \geq 0 \\0 & x<0\end{array}\right.\)

It can be shown that if the times between successive events are independent, each with an exponential distribution with parameter \(\lambda\), then the total time X that elapses before all of the next n events occur has pdf f(x; \(\lambda\), n).a. What is the expected value of X? If the time (in minutes)between arrivals of successive customers is exponentially distributed with \(\lambda\)=.5, how much time can be expected to elapse before the tenth customer arrives?

b. If customer interarrival time is exponentially distributed with \(\lambda\)=.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 \(\leq\) 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 hasa Poisson distribution with parameter \(\lambda\)t to write an expression (involving Poisson probabilities) for the Erlang cdf F(t; \(\lambda\), n) = P(X \(\leq\) t).

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