notes8 - Stat 430/Math468 Notes#8 The Multivariate Normal...

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1 Stat 430/Math468 – Notes #8 Chapter 4 The Multivariate Normal Distribution (Continued) Result: (Conditional distribution) Suppose ~ ( , ) p N X . Make the following partitioning 1 2 = X X X , 1 2 = , and 11 12 21 22 = . Assume that 22 | | 0 . Then the conditional distribution of 1 X given 2 2 = X x is still normal with Mean = 1 1 12 22 2 2 ( ) - + - x and Variance-Covariance Matrix= 1 11 12 22 21 - - . Notation: Suppose the dimension of 1 X is 1 q × . Then the dimension of 2 X is p q - . The result above says that 1 2 2 1 2 11 2 | ( ) ~ ( , ) q N = X X x i i , where 1 1 2 1 12 22 2 2 ( ) - = + - x i and 1 11 2 11 12 22 21 - = - i . Example: (Bivariate Normal) Suppose 2 ~ ( , ) N X . Then 1 2 2 1 2 11 2 | ( ) ~ ( , ) X X x N μ σ = i i , where 12 1 2 1 2 2 22 ( ) x σ μ μ μ σ = + - i and 2 12 11 2 11 22 σ σ σ σ = - i . Result: Suppose ~ ( , ) p N X . Then 1. 1 2 ( )' ( ) ~ p χ - X - X - 2. 1 2 {( )' ( ) ( )} 1 p P χ α α - = - X - X - , where 2 ( ) p χ α is the upper (100 ) α th percentile of the 2 p χ distribution. Proof: is positive definite, then 1 - is also positive definite and 1 1 2 2 1 - - - = . Since 1 2 ( ) ~ - X- Denote 1 2 ( ) - X - by Z with 1 2 ( , ,..., )' p Z Z Z = Z . Then 1 1 2 2 1 ( )' ( ) ( )' ( ) - - - = X - X - X - X- . Note that 1 2 - is also symmetric. So we have 1 ( )' ( ) ' - = = X - X - Z Z
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2 Examples for the bivariate normal Remark: 1. The previous result provides an interpretation for the squared statistical distance.
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