I'm trying to work out a conditioned expectation of a variable that depends on two other via the Law of Total Probability. I've got some intuition on what it should be but there is an important detail that I can't work out, and I need help with it. I've got three discrete random variables $X,Y,Z$, where $Y$ depends on $X,Z$. By the Law of Total Expectation, $E[Y] = E[E[Y|X,Z]]$. Now consider the event $A=X < Z$. My first question is: what is the expression for $E[Y|A]$: is it 1) or 2)?
$$
\begin{align}
1) \qquad
E[Y|A]
&= E[E[Y|X,Z,A]] \\
&= \sum_x \sum_z P(X=x,Z=z|A)E[Y|X=x,Z=z,A] \\
2) \qquad E[Y|A]
&= E[E[Y|X,Z,A]] \\
&= \sum_x \sum_z P(X=x,Z=z,A)E[Y|X=x,Z=z,A]
\end{align}
$$
Note that the only difference between the two expressions is the conditioning of the probability inside the double summation. Maybe it is neither of these? Am I applying the Law of Total Expectation correctly?
I would like to ask another question: am I right when I claim that 3) is correct?
$$
\begin{equation}
3) \qquad E[Y|X=x,Z=z,A] = \sum_y yP(Y=y|X=x,Z=z,A)
\end{equation}
$$
I've seen posts across this site discussing similar topics, but I haven't found anything that could help me. Any help will be very much appreciated.
Best Answer
$E(Y|A) = E(Y1_A) = E(E(Y1_A | X, Z)) = \sum_{x,z}E(Y1_A | X=x, Z=z)P(X=x, Z=z) = \sum_{x<z} E(Y|X=x, Z=z)P(X=x, Z=z)$
As far 3, think about what the event $\{X=x, Z=z, A\}$ really means given that $A = \{X < Z\}$.