A personal computer manufacturer produces both desktop computers and notebook computers. The monthly proportions of customer orders received for desktop and notebook computers that are shipped within 1 week's time can be viewed as the outcome of a bivariate random variable \((X, Y)\) with joint probability density

\(f(x, y)=(2-x-y) I_{[0,1]}(x) I_{[0,1]}(y)\).

a. In a given month, what is the probability that more than 75 percent of notebook computers and 75 percent of desktop computers are shipped within 1 week of ordering?

b. Assuming that an equal number of desktop and notebook computers are ordered in a given month, what is the probability that more than 75 percent of all orders received will be shipped within 1 week?

c. Are the random variables independent?

d. Define the conditional probability that less than 50 percent of the notebook orders are shipped within 1 week, given that \(x\) proportion of the desktop orders are shipped within 1 week (the probability will be a function of the proportion \(x\) ). How does this probability change as \(x\) increases?