Question: 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

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?

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