Lecture 13-2007 - Bivariate and Multivariate Normal Random...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Bivariate and Multivariate Normal Random Variables Lecture XIII I. Bivariate Normal Random Variables A. Definition 5.3.1. The bivariate normal density is defined by 2 22 2 1 , 21 1 exp 2 XY X Y X Y X Y X Y f x y x y x y B. Theorem 5.3.1. Let , have the bivariate normal density. Then the marginal densities X fX and Y fY and the conditional densities f Y X and f X Y are univariate normal densities, and we have X EX , 2 X VX , Y EY , 2 Y VY , , Corr X Y , and | |1 Y YX X Y E Y X X V Y X Note that 2 2 2 2 11 , exp exp 2 2 Y X Y Y X X X f x y y x x ff where 1 f is the density of 2 , XX N and 2 f is the density function of 1 2 2 ,1 Y Y X X Y NX . The proof of the assertions from the theorem can then be seen by: 12 1 f x f f dy f f dy f
Background image of page 1

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

View Full DocumentRight Arrow Icon
AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XIII Professor Charles B. Moss Fall 2007 2 1. This gives us 2 ~, XX XN . Next we have 21 2 1 , | f x y ff f y x f f x f 2. By Theorem 4.4.1 (Law of Iterated Means) ,, X YX E X Y E E X Y (Where the symbol X E denotes the expectation with respect to X ).
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

Lecture 13-2007 - Bivariate and Multivariate Normal Random...

This preview shows document pages 1 - 3. Sign up to view the full document.

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