09_07 - STAT 410 Examples for 09/07/2011 Fall 2011 2.4...

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STAT 410 Examples for 09/07/2011 Fall 2011 2.4 Covariance and Correlation Coefficient Covariance of X and Y σ XY = Cov ( X , Y ) = E [ ( X – μ X ) ( Y – μ Y ) ] = E ( X Y ) μ X μ 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 σ X 2 , σ Y 2 , and σ XY : a) Cov ( 2 X + 3 Y , X – 2 Y ), b) Var ( 2 X + 3 Y ), c) Var ( X – 2 Y ).
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Correlation coefficient of X and Y ρ 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 ρ XY 1; (b) ρ XY is either + 1 or – 1 if and only if X and Y are linear functions of one
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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09_07 - STAT 410 Examples for 09/07/2011 Fall 2011 2.4...

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