Lecture13-2010 - Bivariate and Multivariate Normal Random...

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Bivariate and Multivariate Normal Random Variables: Lecture XIII Charles B. Moss September 21, 2010 I. Bivariate Normal Random Variables A. Defnition 5.3.1. The bivariate normal density is defned by f ( x, y )= 1 2 πσ x σ y q 1 ρ 2 exp 1 2 ± 1 ρ 2 ² " ± x μ x σ x ² 2 + ³ y μ y σ y ´ 2 2 ρ ± x μ x σ x ² ³ y μ y σ y ´ # (1) B. Theorem 5.3.1. Let ( X,Y ) have the bivariate normal density. Then the marginal densities f X ( X )and f Y ( Y ) and the condi- tional densities f ( Y | X f ( X | Y ) are univariate normal den- sities, and we have E [ X ]= μ X ,E [ Y μ Y , V [ X σ 2 X , E[ Y μ Y , V [ Y μ Y ,Corr( ρ ,and Y | X μ Y + ρ σ Y σ X ( X μ X ) V [ Y | X σ 2 Y ± 1 ρ 2 ² (2) Note that 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XIII Fall 2010 f ( x, y )= 1 2 πσ Y q 1 ρ 2 exp 1 2 σ 2 Y ± 1 ρ 2 ² ³ y μ Y ρ σ Y σ X ( x μ X ) 2 1 2 X exp " 1 2 X ( X ) 2 # = f 2 f 1 (3) where f 1 is the density of N ( μ X 2 X )and f 2 is the density func- tion of N ± μ X + ρσ Y σ 1 X ( X μ X ) 2 Y (1 ρ 2 ) ² . The proof of the assertions from the theorem can then be seen by f ( x Z −∞ f 1 f 2 dy f 1 R −∞ f 2 dy = f 1 (4) 1. This gives us x N ( μ X 2 X ). Next, we have f ( y | x f ( x, y ) f ( x ) = f 2 f 1 f 1 = f 2 (5) 2 By Theorem 4.4.1 (Law of Iterated Means) E [
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Lecture13-2010 - Bivariate and Multivariate Normal Random...

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