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If X and Y are independent then they are uncorrelated. It is not true, however, that if they are uncorrelated, they must be independent. For example, if X is uniformly distributed on [−1, 1] and Y = X2 then they are uncorrelated even though X determines Y, and Y restricts X to at most two values.
Moreover, uncorrelatedness is a relation between only two random variables, whereas independence can be a relationship between more than two.