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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = i i , where 12 1 2 1 2 2 22 ( ) x = +- i and 2 12 11 2 11 22 =- i . Result: Suppose ~ ( , ) p N X . Then 1. 1 2 ( )' ( ) ~ p - X - X - 2. 1 2 {( ) ' ( ) ( )} 1 p P - = - X - X - , where 2 ( ) p is the upper (100 ) th percentile of the 2 p distribution. Proof: is positive definite, then 1- is also positive definite and 1 1 2 2 1--- = ....
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online