Independence of Random Variables

Question

I was just wondering:

If the joint distribution of $X$ and $Y$ is independent with $Z$, does it imply that $X$ and $Z$ are independent as well?

Can anyone give a counterexample/proof?

Answer 1

First of all, there's no sense to say that a distribution is independent of another distribution. You should say that the pair of random variables $(X,Y)$ is independent of the random variable $Z$. This is equivalent to say that the random variable $f(X,Y)$ is independent of the random variable $Z$ for every Borelian function $f$. In particular $X$ and $Z$ are independent.