Donna is vacationing in Las Vegas. The amount X (in dollars) she takes to the casino each evening is a random variable with a PDF of the form fX(x) = ax, if 0 ≤ x ≤ 40, 0, otherwise. At the end of each night, the amount Y that she has when leaving the casino is uniformly distributed between zero and twice the amount that she came with.
(a) Determine the joint PDF fX,Y (x, y).
(b) What is the probability that on a given night Donna makes a positive profit at the casino?
(c) Find the PDF of Donna's profit Y - X on a particular night, and also determine its expected value.