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Lecture12-2_Oct_10 - EE 131A Probability Professor Kung Yao...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles M.S. On-Line Engineering Program Lecture 12-2
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UCLA EE131A (KY) 2 Covariance (1) The covariance of rv’s X and Y provides a statistical average relationship between (X- X ) and (Y- Y ) and is defined by Cov(X,Y) = E{(X- X ) (Y- Y )} . Cov(X,Y) = E{XY – X Y – Y X + X Y } = XY X Y Y X + X Y = XY X Y . If X = Y, then Cov(X,Y) = E{(X- X ) 2 } = X 2 = Y 2 . Ex. 1. (Ex. 1, p. 8, Lec12-1). We knew XY = 8/18, X = 4/5, and Y = 8/15. Then Cov(X,Y) = XY X Y = (8/18) – (32/75) .
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UCLA EE131A (KY) 3 Covariance (2) Denote Lemma 1. For any X 1 ,…, X n , we have . (*) Ex. 2. Consider n = 2 in Lemma 1. Then Ex. 3. Consider n = 3 in Lemma 1. Then 2 2 { } {( ) } X X Var X E X 2 1 2 1 1 2 2 1 2 1 2 { } {(( ) ( )) } { } { } 2cov( , ) Var X X E X X Var X Var X X X 1 2 1 , { ... } { } 2 cov( , ) n n j j k j all j k j k Var X X X Var X X X
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