Why does zero correlation not imply independence?

Question

Although independence implies zero correlation, zero correlation does not necessarily imply independence. While I understand the concept, I can't imagine a real world situation with zero correlation that did not also have independence. Can someone please give me an example so I can better understand this phenomenon?

Answer 1

Assuming we have two random variables X and Y and X is a standard normally distribution with mean equals to 0 and standard deviation equals to 1, and Y = X^2. Clearly if we know the value of X then we can compute the value of Y. Hence, X and Y are not independent. HOWEVER, if we compute the correlation coefficient of X and Y, Cor(X,Y) = Cov(X,Y) / [(Var(X) * Var(Y)^0.5], we will have the correlation coefficient Cor(X,Y)=0 because Cov(X,Y) = 0 (Note, there are many of proofs online).