Let Xt the number of customers that have arrived in a bank i

Let Xt the number of customers that have arrived in a bank in the time-interval [0, t]. Suppose that Xt is Poisson with parameter t, so that is the expected number of customers per unit of time. Let be a positive integer and let Y the time of the arrival of the customer. (a) Find the cdf of Y , FY (t) = P(Y t). (b) Show that the pdf of Y follows the Gamma distribution with shape parameter and rate parameter , i.e., that its pdf is fY (t) = (1)! t 1 e t, t > 0 0, t 0 Remark: When = 1, the Gamma distribution reduces to the exponential with parameter . Thus, we have a more general class of distributions that represent “waiting times”. Moreover, the Gamma distribution can be defined more generally for any positive parameter > 0.

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