# Conditional Expectation(s) of Multiple Random Variables

I am having a bit of trouble with one of the problems I am working right now.

The problem is:

"What is $$E\left \{ E\left \{ E\left \{ Z \mid X , Y \right \} \right \} \right \}$$ ?"

During an office hours the professor explained that it would be necessary to use a "probability chain rule" and that you would need to use the following relationship: $$f\left ( x,y,z \right )= f\left ( x \mid y,z \right)f\left( y\mid z \right )f\left(z \right )$$

I am having some trouble seeing how they relate, and also what the actual problem is looking for as well.

Naturally, I am not looking for the answer, but rather a path to the answer. I know I have already been given that path by the professor, but it still does not make much sense to me.

Any help anyone could provide would be very much appreciated!

The notation $$\mathbb E\left \{ \mathbb E\left \{ \mathbb E\left \{ Z \mid X , Y \right \} \right \} \right \}$$ is incorrect, it should be $$\mathbb E [ \overbrace{\mathbb E \{ \underbrace{\mathbb E( Z \mid X , Y )}_\text{function of (X,Y)} \mid X \}}^\text{function of X}]$$ or $$\mathbb E [ \overbrace{\mathbb E \{ \underbrace{\mathbb E( Z \mid X , Y )}_\text{function of (X,Y)} \mid Y \}}^\text{function of Y}]$$
• The inner expectation provides a function of $(X,Y)$, but the following two expectations are not uniquely defined and are missing a supplementary conditioning. That someone has used the notation for several years is not a mathematical argument for validity! Commented Jun 29, 2020 at 13:28
• I interpret the notation differently: the inner expectation is a function of $(X,Y),$ whence the next expectation is the expectation of this bivariate variable--a vector--and finally the outer expectation operator is either nonsensical or should be understood as treating that vector as a constant random variable (making it superfluous).