Chapter 2.6, Problem 18E

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

$f\left(x\right)=\{\begin{array}{l}\frac{3}{2}\left(x^2+y^2\right)\text{ 0 < }x<1\text{ and 0}<y<\text{1}\\ 0\text{ otherwise}\end{array}$

1. a. What is the probability that both machines are in operation for more than half an hour?
2. b. Find the marginal probability density functions f_X(x) and f_Y(y).
3. c. Are X and Y independent? Explain.