I’m kind of having trouble with coming up with an example/thought experiment that could explain this. I tried to look around for a simpler explanation but wasn’t really able to find any. I do know that two independent variables have a covariance of 0 but why is it that the other way around does not work?
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[This picture from wikipedia](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Correlation_examples2.svg/1920px-Correlation_examples2.svg.png) ([article](https://en.wikipedia.org/wiki/Correlation)) explains it nicely.
On the bottom row are a series of 2D distributions with 0 covariance that are very clearly not independent (ie – knowing the position of a point on the x-axis tells you a lot about where that point might lie on the y-axis (except the one on the bottom-right – that one may actually be independent))
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