A painting process consists of two stages. In the first stage, the paint is applied, and in the second stage, a protective coat is added. Let X be the time spent on the first stage, and let Y be the time spent on the second stage. The first stage involves an inspection. If the paint fails the inspection, one must wait three minutes and apply the paint again. After a second application, there is no further inspection. The joint p.d.f. of X and Y is

a. Sketch the region where f (x, y) > 0. It is not exactly a rectangle.
b. Find the marginal p.d.f.€™s of X and Y.
c. Show that X and Y are independent.
This problem does not contradict Theorem 3.5.6. In that theorem the conditions, including that the set where f (x, y) > 0 be rectangular, are sufficient but not necessary.

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if I cr < 3 and 0 < y < 1, 1f 6 < x < 8 and 0 < y < 1, f(x, y) = 0 otherwise.