Andy is vacationing in Las Vegas. The amount X (in Dolars) he takes to casino each evening is a random variable

with a PDF of the form

fX(x)=ax if 0≤x≤23 and

0 otherwise.

At the end of each night, the amount Y that he has when leaving the casino is uniformly distributed between

zero and twice the amount that he came with.

Determine the joint PDF fX,Y(x,y).

fX,Y(8,14)=

What is the probability that on a given night Andy makes a positive profit at the casino?

Find the PDF of Andy's profit Z=Y-X on a particular night.

fZ(0)=

Determine the expected value of Z=Y-X.

E[Z]=

Thank you very much.