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 expectation of quantity sold, as a function of the price of ethanol (i.e., derive the regression curve of \(\mathrm{Q}\) on \(\mathrm{P}\) ).

(b) Graph the expected conditional quantity sold as a function of price. Does the relationship make economic sense?

(c) What is the expected quantity sold if price is \(\$ 2\) ? If price is \(\$ 4\) ?

(d) Derive the expected value of total dollar sales, \(P Q\), of ethanol.