This makes logical sense to me, but I can't seem to prove this. Is this statement true?
Note: X/Y is a ratio here, not conditioning.
2026-03-27 10:12:49.1774606369
With regard to random variables, does $(X/Y)$ independent of $(Y)$ imply that $(X)$ is independent of $(Y)$?
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As pointed out in the comments, that is false. Just make $X=aY$ (extremely dependent variables) and then $X/Y=a$ is independent of $Y$.
It's also false in the other direction. Suppose $X,Y$ are independent, then, letting $Z=X/Y$ (and assuming $Y\ne 0$) then $$P(Z=z | Y=y)= P(X = yz)$$
This will only be independent of $y$ if $X$ is uniform.