3. [32 pts toal] Let X be the time between two successive arrivals at the drive-through window of a local bank. If X has an Gamma distribution with shape and scale parameters = = 1, answer the following:

(a) [7 pts] Show that the distribution of X is equivalent to an exponential distribution with mean parameter = 1. [Hint: remember that the pdf uniquely identifies a distribution]

(b) [5 pts] Find the expectation and variance of the time between two successive arrivals.

(c) [5 pts] Compute the probability P(X <= 4)

(d) [5 pts] Compute the probability P(2 <= X <=5)

(e) [10 pts] Compute the 0.95th quantile of the distribution of X. [Hint: recall that the inverse

transformation of the exponential is the natural logarithm]