# Conditional distribution of two random variables.

Suppose I have two continuous random variables $$X$$ and $$Z$$, $$X$$ and $$Z$$ are not independent. While I know that $$Z \sim \mathcal{N}(0, 1)$$, I have no information for the distribution of $$X$$. Now one of my friend said that the conditional distribution $$p(Z|X)$$ should also be a Gaussian. I am not sure if it's correct or not?

• No this is not correct. I do not know what you mean by $p(Z|X)$, but likely you mean a density $f_{Z|X}(z|x)$ (or perhaps you mean $P[Z\leq z|X=x]$). Anyway, just take $Z$ Gaussian $N(0,1)$ and define $X$ to be uniform over $[0,1]$ if $Z>0$, and $X$ uniform over $[-2,-1]$ if $Z\leq 0$. Then $f_{Z|X}(z|1/2)$ is not a Gaussian PDF since we know it is zero for all $z<0$ (since $X=1/2\implies Z>0$). – Michael Apr 11 at 22:27
• I mean the PDF function of $Z$ given $X$. – lenhhoxung Apr 11 at 22:56