Verification of a Conditional expectation

Question

I need to check if the following formula is correct.

$$ \mathrm{E}(Z \vert X=x)=\sum_{y} \mathrm{E}(Z \vert X=x,Y=y)\mathrm{P}(Y=y \vert X=x) $$

Just as a reference, Law of total expectation states that $\mathrm{E}(Z)= \sum_{x} \mathrm{E}[Z \vert X=x]\mathrm{P}(X=x)$

Any help would be much appreciated :)

Answer 1

It appears that all that was done was to condition $(Z|X)$ itself on $Y$. So as long as the expectation of the twice-conditioned interior is taken against all $Y|X=x$ then that outermost conditioning should drop and we are back where we started. This is sometimes called the law of iterated expectations.