Slides13-2010 - Bivariate and Multivariate Normal Random...

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Bivariate and Multivariate Normal Random Variables: Lecture XIII Charles B. Moss September 21, 2010 Charles B. Moss () Bivariate and Multivariate Normal September 21, 2010 1 / 14
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1 Bivariate Normal Random Variables 2 Multivariate Normal Distribution 3 Conditional Formulations Charles B. Moss () Bivariate and Multivariate Normal September 21, 2010 2 / 14
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Bivariate Normal Random Variables Defnition 5.3.1. The bivariate normal density is defned by f ( x , y )= 1 2 πσ x σ y p 1 ρ 2 exp ± 1 2 ( 1 ρ 2 ) ² ³ x μ x σ x ´ 2 + ³ y μ y σ y ´ 2 2 ρ ³ x μ x σ x ´³ y μ y σ y ´ µ¶ (1) Charles B. Moss () Bivariate and Multivariate Normal September 21, 2010 3 / 14
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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 conditional densities f ( Y | X f ( X | Y ) are univariate normal densities, 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 ( X , Y )= ρ ,and E [ Y | X μ Y + ρ σ Y σ X ( X μ X ) V [ Y | X σ 2 Y ( 1 ρ 2 ) (2) Note that f ( x , y 1 2 πσ Y p 1 ρ 2 exp ± 1 2 σ 2 Y ( 1 ρ 2 ) ² y μ Y ρ σ Y σ X ( x μ X ) ³ 2 ´ 1 2 X exp ² 1 2 X ( x μ X ) 2 ³ = f 2 f 1 (3) Charles B. Moss () Bivariate and Multivariate Normal September 21, 2010 4 / 14
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Where f 1 is the density of N ( μ X 2 X ) and f 2 is the density function of N ( μ X + ρσ Y σ 1 X ( X μ X ) 2 Y ( 1 ρ 2 )) .Thep roo fo fthe assertions from the theorem can then be seen by f ( x )= ± −∞ f 1 f 2 dy f 1 R
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Slides13-2010 - Bivariate and Multivariate Normal Random...

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