# Conditional independence and Conditional Expectation given two random variables

The first question I have is: For conditionally independent (given $$Z$$), random variables $$X$$ and $$Y$$, does $$\mathbb E [X | Y,Z] = \mathbb E [X | Z]$$?

And I also wanted to know if when $$X$$ is independent of $$Y$$ and $$X$$ is independent of $$Z$$, does $$\mathbb E [X | Y,Z] = \mathbb E [X]$$? What about if $$Y$$ and $$Z$$ were also independent (so all $$3$$ random variables are pairwise independent)?

For the first question, I know that if $$\sigma(Y)$$ was independent of the smallest sigma algebra containing $$\sigma(X)$$ and $$\sigma(Z)$$ then it would be true, but conditional independence doesn't imply this. I still think it is true but not sure how to show or prove it. I showed it using the expectation equation with probability densities and that $$p(x|y,z) = \frac{p(xy|z)}{p(y|z)}$$. But I am not sure if this is the correct.

$$\begin{equation} \mathbb E [X | Y,Z] = \int x p(x|y,z)dx = \int x\frac{p(xy|z)}{p(y|z)}dx \\\text{ using conditional independence } = \int x\frac{p(x|z)p(y|z)}{p(y|z)}dx = \int x p(x|z)dx = \mathbb E [X | Z] \end{equation}$$

For the second question, I don't believe it is true since the smallest sigma algebra containing $$\sigma(Y)$$ and $$\sigma(Z)$$ contains more information than each of them alone. And I'm not sure about the expression when $$Y$$ and $$Z$$ are also independent.

Thanks in advance for the help!

• $$X, Y$$ are i.i.d. Bernoulli variables with $$P(X=1) = P(X=0) = 1/2$$.
• $$Z = 1$$ if $$X = Y$$, otherwise $$Z=0$$.
As is well known, these $$3$$ are pairwise independent but not mutually independent - any two of them determine the third.