Stat400Lec14(Ch4.2,5.3) - STAT 400 Chapter 4.2 5.3 Spring...

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Unformatted text preview: STAT 400 Chapter 4.2, 5.3 Spring 2012 4.2 Covariance and Correlation Coefficient Covariance of X and Y XY = Cov ( X , Y ) = E [ ( X – X ) ( Y – Y ) ] = E ( X Y ) – X Y (a) Cov ( X , X ) = Var ( X ) ; (b) Cov ( X , Y ) = Cov ( Y , X ) ; (c) Cov ( a X + b , Y ) = a Cov ( X , Y ) ; (d) Cov ( X + Y , W ) = Cov ( X , W ) + Cov ( Y , W ) . Cov ( a X + b Y , c X + d Y ) = a c Var ( X ) + ( a d + b c ) Cov ( X , Y ) + b d Var ( Y ) . Var ( a X + b Y ) = Cov ( a X + b Y , a X + b Y ) = a 2 Var ( X ) + 2 a b Cov ( X , Y ) + b 2 Var ( Y ) . 0. Find in terms of X 2 , Y 2 , and XY : a) Cov ( 2 X + 3 Y , X – 2 Y ), b) Var ( 2 X + 3 Y ), c) Var ( X – 2 Y ). Correlation coefficient of X and Y XY = Y X XY σ σ σ = , Y Var X Var Y X Cov = Y Y , X X σ μ σ μ Y X E (a) – 1 XY 1; (b) XY is either + 1 or – 1 if and only if X and Y are linear functions of one another. If random variables X and Y are independent, then E ( g ( X ) h ( Y ) ) = E ( g ( X ) ) E ( h ( Y ) ). Cov ( X, Y ) = XY = 0, Corr ( X , Y ) = XY = 0. 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y Recall: E ( X ) = 1.75, E ( Y ) = 0.8, E ( X Y ) = 1.5....
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Stat400Lec14(Ch4.2,5.3) - STAT 400 Chapter 4.2 5.3 Spring...

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