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notes6 - Stat 430/Math468 Notes#6 The Multivariate 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 μ σ has probability density function (PDF) 2 2 1 2 1 2 ( ) 1 ( )( ) ( ) 2 2 2 1 1 ( ) , 2 2 ( ) x x x f x e e x μ μ σ μ σ πσ π σ = = −∞ < < ∞ The shape of PDF curve is unimodal, bell-shaped, and symmetric about μ . Notation: 2 ~ ( , X N ) μ σ . Empirical Rule: ( ) 0.68, ( 2 2 ) 0.95 P X P X μ σ μ σ μ σ μ σ + + 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 1 1 2 2 1 ( )' ( ) 1 2 1 2 2 1 1 1 ( ) ( ,..., ) exp{ ( )' ( )}, 2 (2 ) | | (2 ) | | , 1,..., or p p p i f f x x e x i p π π = = = −∞ < < ∞ = x- μ Σ x- μ x x μ Σ x - μ Σ Σ - ) Notation: , where ~ ( , p N X μ Σ 1 ( ,..., )' p X X = X is a 1 p × random vector. Example: Bivariate Normal with =2. , where p 2 ~ ( , N X μ Σ ) ' 1 2 ( , ) X X = X , 11 12 1 2 21 22 ( , )', σ σ μ μ σ σ = = μ Σ , where ( ) i i E X μ = , ( ), ii i Var X i
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