Lecture 13-2007

# Lecture 13-2007 - Bivariate and Multivariate Normal Random...

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Bivariate and Multivariate Normal Random Variables Lecture XIII I. Bivariate Normal Random Variables A. Definition 5.3.1. The bivariate normal density is defined by 2 22 2 1 , 21 1 exp 2 XY X Y X Y X Y X Y f x y x y x y B. Theorem 5.3.1. Let , have the bivariate normal density. Then the marginal densities X fX and Y fY and the conditional densities f Y X and f X Y are univariate normal densities, and we have X EX , 2 X VX , Y EY , 2 Y VY , , Corr X Y , and | |1 Y YX X Y E Y X X V Y X Note that 2 2 2 2 11 , exp exp 2 2 Y X Y Y X X X f x y y x x ff where 1 f is the density of 2 , XX N and 2 f is the density function of 1 2 2 ,1 Y Y X X Y NX . The proof of the assertions from the theorem can then be seen by: 12 1 f x f f dy f f dy f

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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XIII Professor Charles B. Moss Fall 2007 2 1. This gives us 2 ~, XX XN . Next we have 21 2 1 , | f x y ff f y x f f x f 2. By Theorem 4.4.1 (Law of Iterated Means) ,, X YX E X Y E E X Y (Where the symbol X E denotes the expectation with respect to X ).
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Lecture 13-2007 - Bivariate and Multivariate Normal Random...

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