The weekly average price (in dollars/foot) and total quantity sold (measured in thousands of feet) of copper wire manufactured by the Colton Cable Co. can be viewed as the outcome of the bivariate random variable \((P, Q)\) having the joint density function:

\(f(p, q)=5 p e^{-p q} I_{[1,3]}(p) I_{(0, \infty)}(q)\).

a. What is the probability that total dollar sales in a week will be less than \(\$ 2,000\) ?

b. Find the marginal density of price. What is the probability that price will exceed \(\$ .25 / \mathrm{ft}\) ?

c. Find the conditional density of quantity, given price \(=.20\). What is the probability that \(>5,000 \mathrm{ft}\) of cable will be sold in a given week?

d. Find the conditional density of quantity, given price \(=.10\). What is the probability that \(>5,000 \mathrm{ft}\) of cable will be sold in a given week? Compare this result to your answer in (c). Does this make economic sense? Explain.