notes8 - 1 Stat 430/Math468 – Notes#8 Chapter 4 The...

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Unformatted text preview: 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 | | . 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--- = ....
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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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notes8 - 1 Stat 430/Math468 – Notes#8 Chapter 4 The...

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