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Unformatted text preview: lent expression can be stated: Cov(X , Y ) = E[(X − μ X ) (Y − μ Y )]
= E[(X Y − X μ Y − μ X Y + μ X μ Y )] = E(XY ) − μ X μ Y − μ X μ Y + μ X μ Y = E(XY ) − μ X μ Y
where E(X Y ) = ∑ ∑ x y PX , Y (x , y ) xy 30 Chapter 5 If the random variables X and Y are independent then: E(X Y ) = ∑ ∑ x y PX , Y (x , y )
xy = ∑ ∑ x y PX (x ) PY (y )
xy = [∑ x PX (x )] [∑ y PY (y )]
x y = μX μY
It follows that independence gives: Cov(X , Y ) = E(XY ) − μ X μ Y = 0 However, if zero covariance is established, this does not guarantee that the random variables are independent. Covariance is designed to measure the possibility of a linear relationship. Nonlinear relationships between variables may give dependencies even though the covariance is zero. Also note that, in general, for random variables with non‐zero covariance: 31 E(XY ) ≠ E(X ) E(Y ) Chapter 5 Covariance gives an indication of the sign (positive or negative) of a linear relationship between...
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This note was uploaded on 02/06/2014 for the course ECON ECON 325 taught by Professor Whistler during the Spring '10 term at The University of British Columbia.
- Spring '10