Expected Value and Conditional Probability

[retspool]

So i a need to find E[XY], expected value of XY

But the process of finding E[X] includes a long and tideous integral which i am trying to avoid.
So computing E[XY] using its formula is also something i am trying to avoid.

But could i use this identity?

E[XY] = E[E(XY/Y)] = E[Y[E(X/Y)]]

Since I've already found E[X/Y] = Y, it gives me E[Y.Y] = E[Y^2].

Is the identity true?

 

[retspool]

I deduced the identity from the given formula

E[X] = E(E[XY/Y])