Thread: Expectation

I have attached the question on this thread.

I have managed to attain the following:





Yet I cannot obtain the fact that



I'm not sure how to derive this using my results.

Hello,

Consider that E[XY]=E[E[XY|X]]

But E[XY|X]=X*E[Y|X].

Since you know the pdf of Y|X, you'll easily get the expectation. Then you'll get E[something with respect to X], which is again easy because you have the pdf of X.

And is follows that:



and then we just take the expectation



Am I correct?

Yes, it's correct.

A few remarks :
- are you aware of the properties of the conditional expectation that have been used ?
- in the pdf of X and of Y|X, you should respectively add that it's for x>0 and y>0