A production facility contains two machines that are used to rework items that are initially defective. Let X be the number of hours that the first machine is in use, and let Y be the number of hours that the second machine is in use, on a randomly chosen day. Assume that X and Y have joint probability density function given by.

a. What is the probability that both machines are in operation for more than half an hour?

b. Find the marginal probability density functions f_X (x) and f_Y (y).

c. Are X and Y independent? Explain.

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