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 conditional-on-p PDF for quantity sold.

b. What is the probability that quantity sold exceeds 50,000 gal if price \(=\$ 2\). What is the probability that quantity sold exceeds 50,000 gal if price \(=\$ 4\). Does this make economic sense?

c. What is the probability that quantity sold exceeds 50,000 gal if price is greater than or equal to \(\$ 3.00\) ?