Lecture 09-2007

Lecture 09-2007 - MOMENTS OF MORE THAN ONE RANDOM VARIABLE...

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MOMENTS OF MORE THAN ONE RANDOM VARIABLE Lecture IX

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Covariance and Correlation Fall 2007 Fall 2007 Lecture IX Lecture IX 2 Definition 4.3.1: ( 29 [ ] ( 29 [ ] ( 29 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Y E X E XY E Y E X E Y E X E Y E X E XY E Y E X E Y X E Y XE XY E Y E Y X E X E Y X Cov - = + - - = + - - = - - = ,
Fall 2007 Fall 2007 Lecture IX Lecture IX 3 Note that this is simply a generalization of the standard variance formulation. Specifically, letting Y -> X yields: ( 29 [ ] [ ] [ ] [ ] [ ] ( 29 2 2 X E X E X E X E XX E XX Cov - = - =

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Fall 2007 Fall 2007 Lecture IX Lecture IX 4 From a sample perspective, we have: ( 29 ( 29 = = = = n t t t n t t y x n Y X Cov x n X V 1 1 2 1 , 1
Fall 2007 Fall 2007 Lecture IX Lecture IX 5 Together the variance and covariance matrices are typically written as a variance matrix: ( 29 ( 29 ( 29 ( 29 = = Σ yy yx xy xx Y V Y X Cov Y X Cov X V σ , , ( 29 ( 29 X Y Cov Y X Cov yx xy , , = = =

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Sample Variance Matrix Fall 2007 Fall 2007 Lecture IX Lecture IX 6 Substituting the sample measures into the variance matrix yields = = = = = = = = = = = n t t t n t t t n t t t n t t t n t t t n t t t n t t t n t t t yy yx xy xx y y x y y x x x n y y n x y n y x n x x n s s s s S 1 1 1 1 1 1 1 1 1 1 1 1 1
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Lecture 09-2007 - MOMENTS OF MORE THAN ONE RANDOM VARIABLE...

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