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If random variables X and Y are correlated, but Y and Z are iid, is X and Z always independent? I am able to prove they are independent for specific types of structure on X and Y (for example if $X=Y+U$ where $U$ is iid), but is this true in general. Not sure why I am having such a difficult time proving the general case?

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A similar question is asked at, which has several useful answers. – whuber Jun 20 '14 at 15:23
Imagine X, Y, Z are a bunch of 3 vectors (arrows) in 3d space going from the same point. "Independent" means 90 degree angle. Any other angle is "correlated". Visualize the possibilities... – ttnphns Jun 20 '14 at 15:24
1. You can't prove a negative. 2. It can be disproven and has been by @heropup's counterexample. – Aaron Hall Jun 20 '14 at 19:01
up vote 13 down vote accepted


Counterexample: Suppose $Y, Z \sim {\rm Bernoulli}(0.5)$ are IID. Now let $X = YZ$. $X$ clearly is not independent of either $Y$ or $Z$.

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