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

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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 $Y$.

Less trivial examples can be made too.

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