The daily price/gallon and quantity sold (measured in millions of gallons) of a lubricant sold on the wholesale spot market of a major commodity exchange is the outcome of a bivariate random variable \((P, Q)\) having the joint probability density function

\(f(p, q)=2 p e^{-p q} I_{[.5,1]}(p) I_{(0, \infty)}(q)\).

(a) Define the regression curve of \(q\) on \(p\).

(b) Graph the regression curve that you have defined in (a).

(c) What is the expected value of the quantity of lubricant sold, given that price is equal to \(\$ .75\) per gallon?

(d) What is the expected value of total dollar sales of lubricant on a given day?