Problem 5 [3+3+4 = 10 points].
A company pays a fine if the time to process requests exceeds 7 seconds. Processing a request consists
of two tasks: (a) retrieving the file, which takes an amount of time X that is exponentially distributed
with mean 5 seconds, and (b) processing the file, which takes an amount of time Y that is independent
of X and uniformly distributed between 1 second and 3 seconds. Given that the mean time to process a
request is 7 seconds, the company views the fine as unfair, because it will have to pay the fine on half of
its requests. In this problem you will investigate whether the company is correct by computing the actual
fraction of requests for which the fine will have to be paid.
1) Find the joint PDF $f_{X,Y}(x, y)$ of X and Y.
2) Sketch the region for which $f_{X,Y}(x,y)$ is positive, and the subregion over which the company will
have to pay the fine.
3) Carry out the integration and compute $P(X+Y > 7)$, and draw a conclusion on whether the company
was right or wrong.
