62 independence and correlation if x and y are

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Independence and Correlation If X and Y are independent random variables with E X 2 , E Y 2 then they are always uncorrelated: Cov X , Y 0. We already covered a more general version of this finding. In fact, for any functions g X and h Y , Cov g X , h Y  0. Why? Under independence we showed E g X h Y  E g X  E g Y  and so Cov g X , h Y  E g X h Y  E g X  E g Y  0 63
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Independence is a much stronger requirement than uncorrelatedness. For example, suppose X has a symmetric distribution about its mean of zero. Let Y X 2 . Then X and Y are clearly not independent. For example, P Y 1,| X | 1 P X 2 1,| X | 1 P | X | 1 P X 2 1 P | X | 1 P Y 1 P | X | 1 unless P | X | 1 0 or 1. 64
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Interestingly, X and Y X 2 are uncorrelated: Cov X , X 2 E X X 2 E X 3 0 by symmetry. Even though Y is a deterministic function of X – once we know X , we always know Y Y is not linearly related to X . Correlation is often described as a measure of linear association. For many purposes it is suitable, but it can miss more complicated relationships. 65
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As it turns out, if g X and h Y are uncorrelated for all functions g  and h  with finite second moments, then X and Y must be independent. This provides an idea of how much stronger independence between X and Y is compared with just saying X and Y are uncorrelated. 66
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Variance - Covariance Matrix Let X be an m 1 random vector with E X j 2 . We want to define the variance matrix of X , sometimes called the variance - covariance matrix of X or (less preferred) the covariance matrix of X . Let j 2 Var X j , j 1,2,..., m ij Cov X i , X j , i j so we have m (possibly) different variances and m m 1 /2 (possibly) distinct covariances. 67
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Arrange these in an m m matrix and call this Var X : Var X 1 2 12 1 m 12 2 2 2 m 1 m 2 m m 2 Sometimes we write X Var X or even Var X . Note that ij ji has been imposed. This means that Var X is a symmetric matrix: X X . 68
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Often it is convenient to obtain Var X from a matrix expectation. Note that X X 1 1 X 2 2 X m m and so 69
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X  X X 1 1 X 2 2 X m m X 1 1 X 2 2 X m m X 1 1 2 X 1 1  X m m X 1 1  X 2 2 X 2 2  X m m X 1 1  X m m X m m 2 70
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It follows that Var X E  X 
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