# A question about conditional independence

Suppose we have three random variables $X,Y,Z$.

1) If $X$ and $Y$ are independent, are they still independent given $Z=z$?

2) If $X$ and $Y$ are independent given $Z=z$, are $X$ and $Y$ independent?

If true, please give a proof. If false, please give a counterexample. Thanks a lot for any reply.

-

1) Suppose that $Z = z$ if and only if $Y = X$. Then $X$ and $Y$ are no longer independent.
2) Suppose that $X$ always equals $Y$ and $X = Y = c$ (for some constant $c$) in the case where $Z = z$. Then $X$ and $Y$ are (vacuously) independent given $Z = z$, but they are not independent in the general case.
If $X$ always equals $Y$, then knowledge of one determines the other. How are they independent? –  alancalvitti Feb 14 '13 at 5:03
1) A case where this happens is if you define $Z = X - Y$ and $z = 0$. 2) A case where this happens is if you define $X = Y = Z$ and $c = z$. –  Joe Z. Feb 14 '13 at 16:40