# If X is independent of Y given Z, is X/Z independent of Y?

When is the following true? If $X$ is independent of $Y$ given $Z$, $\frac{X}{Z}$ is independent of Y.

There is no reason to expect such a result. For instance, take $$X$$ a positive constant with probability 1, and $$Y=Z$$. As constants are independent of everything (even themselves!), certainly $$X$$ and $$Y$$ are independent given $$Z$$. But $$\frac{X}{Z}$$, which is proportional to $$Y^{-1}=Z^{-1}$$, will not be independent of $$Y$$.