The daily price and quantity sold of wheat in a Northwestern market during the first month of the marketing year is the outcome of a bivariate random variable \((P, Q)\) having the probability density function

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

where \(p\) is measured in \(\$ /\) bushel, and \(q\) is measured in units of 100,000 bushels.

(a) Define the conditional expectation of quantity sold as a function of price, i.e., define \(\mathrm{E}(\mathrm{Q} \mid p)\) (the regression curve of \(Q\) on \(P\) ).

(b) Graph the regression curve you derived in (a). Calculate the values of \(\mathrm{E}(\mathrm{Q} \mid p=3.50)\) and \(\mathrm{E}(\mathrm{Q} \mid p=4.50)\).