The daily price, \(p\), and daily quantity sold, \(q\), of pink salmon produced by the AJAX Fish Packing Co. can be represented by outcomes of the bivariate random variable \((P, Q)\) with density function

\(f(p, q)=5 p e^{-p q} I_{[.2,4]}(p) I_{(0, \infty)}(q)\)

where \(p\) is measured in dollars, and \(q\) is measured in 1,000 's of pounds.

(a) Derive the density function for \(R=P Q\), where outcomes of \(R\) represent daily revenue from fish sales. (Hint: Define \(W=P\) as an auxiliary random variable, and use the change of variables approach).

(b) What is the expected value of daily revenue? What is the probability that daily revenue exceeds \(\$ 300\) ?