The law of total expectation states that:
E(X) = E[E(X|Y)] and E[g(X)] = E[E(g(X)|Y)]
1) Now, is it correct to say that E(XY)=E[E(XY|Y)] ? I don't think the above law applies here, because in the law of total expectation the red part has to be a function of X alone (in particular, it cannot depend on Y), but here we have XY which is NOT a function of X alone. It is a function of both X and Y. Is that OK?
2) How about E[X h(Y)]=E[E(X h(Y)|Y)]? Is this a correct statement?
So I am really confused...and I would appreciate if anyone can help
[note: also under discussion in talk stats forum]