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Unformatted text preview: Bivariate and Multivariate Normal Random Variables Lecture XIII Bivariate Normal Random Variables Definition 5.3.1. The bivariate normal density is defined by ( 29 ( 29    +  = Y Y X X Y Y X X Y X y x y x y x f 2 1 2 1 exp 1 2 1 , 2 2 2 2 Theorem 5.3.1. Let ( X , Y ) have the bivariate normal density. Then the marginal densities f ( x ) and f ( y ) and the conditional densities f ( y  x ) and f ( x  y ) are univariate normal densities, and we have E [ X ]= X , V [ X ]= X 2 , E [ Y ]= Y , V [ Y ]= Y 2 , Corr ( X , Y )= , and [ ] ( 29 [ ] ( 29 2 2   1 Y Y X X Y E Y X X V Y X = +  =  ( 29 ( 29 ( 29 ( 29 1 2 2 2 2 2 2 2 2 1 exp 2 1 1 2 1 exp 1 2 1 , f f x x y y x f X X X X X Y Y Y Y =   = where f 1 is the density of N ( X , X 2 ) and f 2 is the density function of...
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This note was uploaded on 11/08/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.
 Fall '09
 CARRIKER

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