[alThor]

Hints only:

You have (a) right.

There is an easier way to do (b). What does common sense tell you the answer is? (If you can't figure out the easier way, you could still use your method and figure out E(XY_bar * YX_bar) using integrals.)

(c) is more difficult. To get a common-sense answer, suppose p isn't fixed, but suppose you fix X first, instead. Then randomly draw Y. Given Y, finally, draw p randomly. Do you think p will end up being in XY or YX? How likely are those two outcomes, relative to each other? After you figure that out, draw the density function 'f'. Calculate the cumulative 'F' only afterwards.

alThor