02_08ans - STAT 410 Examples for 02/08/2008 Spring 2008...

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Unformatted text preview: STAT 410 Examples for 02/08/2008 Spring 2008 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 ) . 1. Find in terms of X 2 , Y 2 , and 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 ). 2. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y x 0 1 2 p X ( x ) Recall: 1 0.15 0.15 0 0.30 E ( X ) = 1.7 2 0.15 0.35 0.20 0.70 E ( Y ) = 0.9 p Y ( y ) 0.30 0.50 0.20 E ( X Y ) = 1.65 Find Cov ( X , Y ) = XY . Cov ( X , Y ) = 1.65 1.7 0.9 = 0.12....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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02_08ans - STAT 410 Examples for 02/08/2008 Spring 2008...

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