# Conditional independence of two random variables given a third random variable

I have three random variables $$(X, Y, Z)$$. The variables $$X$$ and $$Y$$ are conditionally independent given $$Z$$.

Will it be correct if I assume $$P(X | Z) = P(X)$$ or $$P(X | Y) = P(X)$$ ? If not, then why? Can you give a simple example to explain why the above assumption will not be correct?

Let $$X:=\epsilon_1+Z$$ and $$Y:=\epsilon_2+Z$$, where $$\epsilon_1$$ and $$\epsilon_2$$ are independent $$N(0,1)$$ (independent of $$Z$$) and $$\mathsf{P}(Z=1)=\mathsf{P}(Z=-1)=1/2$$. $$\mathsf{P}(X\le 0\mid Z)=\mathsf{P}(\epsilon_1+Z\le 0\mid Z)=\Phi(-Z).$$ But $$\mathsf{P}(X\le 0)=1/2$$.
Conditional independence means that for all Borel sets $$A$$ and $$B$$, $$\mathsf{P}(X\in A, Y\in B\mid Z)=\mathsf{P}(X\in A\mid Z)\mathsf{P}(Y\in B\mid Z).$$
• Is $\Phi(Z)$ a cumulative probability function? And, is $Z$ symmetric about 0 ? – user10853036 Feb 5 '19 at 22:08
• Yes, $\Phi$ is the cdf of $N(0,1)$. – d.k.o. Feb 5 '19 at 22:31