The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable \((P, Q)\) having the following probability model \((R(P, Q), f(p, q))\) :

\(f(p, q) \begin{cases}.5 p e^{-p q} \text { for } & (p, q) \in R(P, Q)=[2,4] \times[0, \infty) \\ 0 & \text { elsewhere }\end{cases}\)

where price is measured in dollars and quantity is measured in 100,000 gal units (e.g., \(q=2\) means 200,000 gal were sold).

a. Derive the marginal probability density of price. Use it to determine the probability that price will exceed \(\$ 3\).

b. Derive the marginal cumulative distribution function for price. Use it to verify your answer to part

(a) above.

c. Derive the marginal probability density of quantity. Use it to determine the probability that quantity sold will be less than \(\$ 500,000\) gal.

d. Let the random variable \(D=P Q\) denote the daily total dollar sales of ethanol during the summer months. What is the probability that daily total dollar sales will exceed \(\$ 300,000\) ?

e. Are \(P\) and \(Q\) independent random variables?