Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf:
f(x) = Ax^2 for 0 < x < 10
where A is a constant that you need to determine. He returns with an amount of money Y that is uniformly distributed between 0 and 3X. Find
- E[Y] w/out finding the distribution for Y (i.e. using E[E(Y|X)])
- The pdf for Y
- E[Y] using the distribution for Y. Compare with your result of part a.
- The prob he returns w/ at least 5 dollars
- If he returns w/ 5 dollars, what is the prob he left with at least 7 dollars
- If he return w/ 5 dollars, what is the expected value of the amount of money with which he left?
I am having a hard time finding the joint pdf for this situation. I know f(x) = 3x^2/1000 for 0 < x < 10, but I keep getting 1/3X for the distribution of Y, which doesn't seem right. Any hints on how to form the joint pdf here?