The average price and total quantity sold of an economy brand of ballpoint pen in a large western retail market during a given sales period is represented by the outcome of a bivariate random variable having a probability density function

\(f(p, s)=10 p e^{-p s} I_{[\cdot 10, .20]}(p) I_{(0, \infty)}(s)\)

where \(p\) is the average price, in dollars, of a single pen and \(s\) is total quantity sold, measured in 10,000-pen units.

(a) Define the regression curve of \(S\) on \(P\).

(b) What is the expected quantity of pens sold, given that price is equal to \(\$ 0.12\) ? (You may use the regression curve if you wish.)

(c) What is the expected value of total revenue from the sale of ball point pens during the given sales period, i.e., what is \(\mathrm{E}(P S)\) ?