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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Independence is an extremely strong property. It means that knowing any information about one variable do not give you any information about another variable. In particular, that means *any* 2 functions on each variable are uncorrelated, their covariance is 0.
Covariance 0 merely means that that they 2 variables are linearly uncorrelated, it says nothing about other non-linear functions of these variables. For example, from Cov(X,Y)=0 you don’t know anything about Cov(X^2 , Y^2 ). For example, let’s say X^2 =Y^2 =Z, where Z is a random positive number with positive variance, there are 50% chance of each variable being positive and negative independently for any given value of Z, and their variance is positive. Then X and Y are not independent, since knowing one variable tell you a lot about the other variable, and in fact Cov(X^2 ,Y^2 ) is positive. But Cov(X,Y)=0 because knowing one variable does not tell you whether the other is positive or negative.
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