Question: 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

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.

3 + y?) z(x² 0<x < 1 and 0 < y < 1 f (x) = x). otherwise

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.

3 + y?) z(x 0

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