I have a few examples I could provide, here is one. Let X = Uniform(-1,1) and X = $Y^2$. Here we have clearly E(X) = 0 and then E(Y) = $\frac{1}{3}$.
Cov(X,Y) = E(X*Y) - E(X) * E(Y) = E($X^3$) - 0 = E($X^3$) = 0. But obviously X and Y are not independent as Y is quite literally valued from X. What does this really mean? What is the interpretation of "Covariance" ? Any insight into some theory behind statistics would be greatly appreciated. :)
Oskar