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Unformatted text preview: Stat 351 Fall 2007 Assignment #7 Solutions 1 . If X 1 , X 2 are independent N (0 , 1) random variables, then by Definition I, Y 1 = X 1 3 X 2 + 2 is normal with mean E ( Y 1 ) = E ( X 1 ) 3 E ( X 2 ) + 2 = 2 and variance var( Y 1 ) = var( X 1 3 X 2 + 2) = var( X 1 ) + 9 var( X 2 ) 6 cov( X 1 ,X 2 ) = 1 + 9 0 = 10, and Y 2 = 2 X 1 X 2 1 is normal with mean E ( Y 2 ) = 2 E ( X 1 ) E ( X 2 ) 1 = 1 and variance var( Y 2 ) = var(2 X 1 X 2 1) = 4 var( X 1 ) + var( X 2 ) 4 cov( X 1 ,X 2 ) = 4 + 1 0 = 5. Since cov( Y 1 ,Y 2 ) = cov( X 1 3 X 2 + 2 , 2 X 1 X 2 1) = 2 var( X 1 ) 7 cov( X 1 ,X 2 )+3var( X 2 ) = 2 0+3 = 5, we conclude that Y = ( Y 1 ,Y 2 ) is multivariate normal N ( , ) where = 2 1 and = 10 5 5 5 . 2 . Let B = 1 0 1 0 2 0 so that Y = B X . By Theorem 3.1, Y is MVN with mean B = 1 0 1 0 2 0 3 4 3 = 8 and covariance matrix B B = 1 0 1 0 2 0 2 1 3 1 4 2 3 2 8 1 0 0 2 1 0 = 16 2 2 16 . 3 . Let B = 1 1 2 1 2 2 0 3 so that Y = B X . By Theorem 3.1, Y is MVN with mean B = 1 1 2 1 2 2 0 3 = and covariance matrix...
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 Fall '08
 MichaelKozdron
 Probability, Variance

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