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I understand that two independent random variables are by definition uncorrelated as their covariance is equivalent to 0:

$Cov(x,y) = E(xy)- E(x)E(y)$

$E(x)*E(y) = E(xy)$, when x and y are two random independent variables.

Therefore, $Cov(x,y) = 0$.

However, I am having trouble understanding if two random variables, X and Y, are uncorrelated, it does not necessarily mean they are independent.

Could someone also give me a real world example of when two random variables are neither independent nor casually connected?

I believe it will help me understand this concept better.