Following Example 4.21, let X and Y denote the arrival times of the first two calls at a telephone switch. The joint PDF of X and Y is

What is the PDF of W = Y − X?

Example 4.21

At noon on a weekday, we begin recording new call attempts at a telephone switch. Let X denote the arrival time of the first call, as measured by the number of seconds after noon. Let Y denote the arrival time of the second call. In the most common model used in the telephone industry, X and Y are continuous random variables with joint PDF

Where λ > 0 calls/second is the average arrival rate of telephone calls. Find the marginal PDFs f_X (x) and f_Y (y) and the conditional PDFs f _X|Y (x|y) and f_{Y |X} (y|x).

 

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