23. ACE Rentals, a car-rental company, rents three types of cars: compacts, mid-size sedans, and large luxury cars.

Let (x1, x2, x3) represent the number of compacts, mid-size sedans, and luxury cars, respectively, that ACE rents per day. Let the sample space for the possible outcomes of (X1, X2, X3) be given by S = f (x1;x2;x3Þ :x1;x2, and x3 2 (0,1,2,3)g

(ACE has an inventory of nine cars, evenly distributed among the three types of cars).

The discrete PDF associated with (X1, X2, X3) is given by fðx1; x2; x3Þ ¼ :004 3ð Þ þ 2x1 þ x2

ð Þ 1 þ x3 P

3 i¼1 If0;1;2;3g ðxiÞ:

The compact car rents for $20/day, the mid-size sedan rents for $30/day, and the luxury car rents for $60/day.

a. Derive the marginal density function for X3. What is the probability that all three luxury cars are rented on a given day?

b. Derive the marginal density function for (X1, X2).

What is the probability of more than one compact and more than one mid-size sedan being rented on a given day?

c. Derive the conditional density function for X1, given x2  2. What is the probability of renting no more than one compact care, given that two or more midsize sedans are rented?

d. Are X1, X2, and X3 jointly independent random variables? Why or why not? Is (X1, X2) independent of X3?

e. Derive the conditional density function for (X1, X2), given that x3 ¼ 0. What is the probability of renting more than one compact and more than one mid-size sedan given that no luxury cars are rented?

f. If it costs $150/day to operate ACE Rentals, define a random variable that represents the daily profit made by the company. Define an appropriate density function for this random variable. What is the probability that ACE Rentals makes a positive daily profit on a given day?