09_22ans - STAT 410 Examples for 09/22/2008 Fall 2008 2.4...

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Unformatted text preview: STAT 410 Examples for 09/22/2008 Fall 2008 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 ) . 1. Find in terms of σ X 2 , σ Y 2 , and σ XY : a) Cov ( 2 X + 3 Y , X – 2 Y ), Cov ( 2 X + 3 Y , X – 2 Y ) = 2 Var ( X ) – Cov ( X , Y ) – 6 Var ( Y ). b) Var ( 2 X + 3 Y ), Var ( 2 X + 3 Y ) = Cov ( 2 X + 3 Y , 2 X + 3 Y ) = 4 Var ( X ) + 12 Cov ( X , Y ) + 9 Var ( Y ). c) Var ( X – 2 Y ). Var ( X – 2 Y ) = Cov ( X – 2 Y , X – 2 Y ) = Var ( X ) – 4 Cov ( X , Y ) + 4 Var ( Y ). Correlation coefficient of X and Y ρ XY = Y X XY & & & = ( ) ( ) ( ) , Y Var X Var Y X Cov ⋅ = & & ¡ ¢ £ £ ¤ ¥ ¦ ¦ § ¨ © © ª « ¦ ¦ § ¨ © © ª «-- Y Y , X X & ¡ & ¡ Y X Cov (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 another....
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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.

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09_22ans - STAT 410 Examples for 09/22/2008 Fall 2008 2.4...

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