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Zero Covariance Does Not Necessarily Imply Independence. Let X = −2, −1, 0, 1, 2 with Pr[X = x] = 1/5. Assume a perfect quadratic relationship between Y and X , namely Y = X 2. Show that cov(X, Y ) = E(X 3) = 0. Deduce that ρXY = correlation (X, Y ) = 0. The simple correlation coef- ﬁcient ρXY measures the strength of the linear relationship between X and Y . For this example, it is zero even though there is a perfect nonlinear relationship between X and Y . This is also an
example of the fact that if ρXY = 0, then X and Y are not necessarily independent. ρxy = 0 is a necessary but not suﬃcient condition for X and Y to be independent. The converse, however, is true, i.e., if X and Y are independent, then ρXY = 0, see problem 2.
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