The daily price, \(p\), and quantity sold, \(q\), of ground beef produced by the Red Meat Co. can be represented by outcomes of the bivariate random variable \((P, Q)\) having bivariate density function

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

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

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

(b) What is the expected value of daily revenue? What is the probability that daily revenue exceeds \(\$ 1,000\) ?