Law of Iterated Expectations $\operatorname{E}\left[XYZ\right]=\operatorname{E}\left[\operatorname{E}\left[XYZ \mid X,Y\right]\right]$

Question

my doubt is actually pretty simple.

Considerer three random variables $X,Y,Z$. I wonder if I can use the LIE to do this: $$\operatorname{E}\left[XYZ\right]=\operatorname{E}\left[\operatorname{E}\left[XYZ \mid X,Y\right]\right]$$

Or the only way to condition the expected value for two variable is to use the other version of LIE, for example: $$E \left[ E \left(Y\mid X,Z \right) \mid X \right] =E \left[Y \mid X \right]$$

Thank you very much.

Answer 1

Please see if this helps?

\begin{align*} \mathbb{E}[XYZ] &= \sum\limits_{x_i,y_j,z_k} x_i\cdot y_j\cdot z_k\cdot\mathbb{P}(X=x_i,Y=y_j,Z=z_k) \\ &= \sum\limits_{x_i,y_j,z_k} x_i\cdot y_j\cdot z_k\cdot\mathbb{P}(Z=z_k|X=x_i,Y=y_j)\mathbb{P}(X=x_i,Y=y_j) \\ &= \sum\limits_{x_i,y_j} \mathbb{E}_{Z|X,Y}[XYZ|X=x_i,Y=y_j] \cdot \mathbb{P}(X=x_i,Y=y_j) \\ &= \mathbb{E}_{X,Y}\left[\mathbb{E}_{Z|X,Y}[XYZ|X=x_i,Y=y_j]\right] \end{align*}