Notes6 - Stat 430/Math468 Notes#6 The Multivariate Normal Distribution Chapter 4 Univariate Normal Distribution Recall the univariate normal

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Stat 430/Math468 – Notes #6 Chapter 4 The Multivariate Normal Distribution Univariate Normal Distribution Recall the univariate normal distribution 2 (, ) N μσ has probability density function (PDF) 2 21 2 1 2 () 1 ( ) 2 2 2 11 , 2 2( ) x xx fx e e x μ σ πσ −− == < < The shape of PDF curve is unimodal, bell-shaped, and symmetric about . Notation: 2 ~( , XN ) . Empirical Rule: ( ) 0.68, ( 2 2 ) 0.95 PX P X σμ −≤ ≤+ ≈ − ≤ ≤+ ≈ Multivariate Normal Distribution PDF: The probability density function (PDF) for a p-dimensional normal distribution with mean vector μ and variance-covariance matrix is Σ 1 22 1 '() 1 2 1 1 ( ) ( ,. .., ) exp{ ( )' ( )}, 2 (2 ) | | (2 ) | | ,1 , . . . , or p pp i ff x x e xi p ππ = −∞ < < ∞ = x- μΣ μ μ ΣΣ - ) Notation: , where , p N X 1 ( ,..., p XX = X is a 1 p × random vector. Example: Bivariate Normal with =2. , where p 2 , N X ) ' 12 X X = X , 11 12 21 22 ' , μμ σσ ⎛⎞ ⎜⎟ ⎝⎠ , where ( ) ii EX = ,( ) , ii i Var X i 1 , 2 = = . 12 21 1 2 Cov X X . Show that the PDF of can be written as X 2 2 12 2 2 12 11 22 11 22 11 22 12 e x p 2 , 2(1 ) 2( 1 ) , 2 , i x x fxx ρ πσσ
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This note was uploaded on 03/07/2010 for the course STAT 430 taught by Professor Hua during the Spring '10 term at University of Illinois at Urbana–Champaign.

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Notes6 - Stat 430/Math468 Notes#6 The Multivariate Normal Distribution Chapter 4 Univariate Normal Distribution Recall the univariate normal

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