I have attached the question on this thread.

I have managed to attain the following:

$\displaystyle P_x(x)=\alpha^2xe^{-\alpha x}$

$\displaystyle P_{y|x}(y|x)=xe^{-xy}$

Yet I cannot obtain the fact that

$\displaystyle E[XY]=1$

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