ACE Rentals, a car-rental company, rents three types of cars: compacts, mid-size sedans, and large luxury cars. Let \(\left(x_{1}, x_{2}, x_{3}ight)\) 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 \(\left(X_{1}ight.\), \(X_{2}, X_{3}\) ) be given by

\(S=\left\{\left(x_{1}, x_{2}, x_{3}ight): x_{1}, x_{2}ight.\), and \(\left.x_{3} \in(0,1,2,3)ight\}\)

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

The discrete PDF associated with \(\left(X_{1}, X_{2}, X_{3}ight)\) is given by

\(f\left(x_{1}, x_{2}, x_{3}ight)=\left[\frac{.004\left(3+2 x_{1}+x_{2}ight)}{\left(1+x_{3}ight)}ight] \prod_{i=1}^{3} I_{\{0,1,2,3\}}\left(x_{i}ight)\).

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 \(X_{3}\). What is the probability that all three luxury cars are rented on a given day?

b. Derive the marginal density function for \(\left(X_{1}, X_{2}ight)\). 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 \(X_{1}\), given \(x_{2} \leq 2\). What is the probability of renting no more than one compact care, given that two or more midsize sedans are rented?

d. Are \(X_{1}, X_{2}\), and \(X_{3}\) jointly independent random variables? Why or why not? Is \(\left(X_{1}, X_{2}ight)\) independent of \(X_{3}\) ?

e. Derive the conditional density function for \(\left(X_{1}, X_{2}ight)\), given that \(x_{3}=0\). 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?