solutions07 - Stat 351 Fall 2007 Assignment #7 Solutions 1...

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: Stat 351 Fall 2007 Assignment #7 Solutions 1 . If X 1 , X 2 are independent N (0 , 1) random variables, then by Definition I, Y 1 = X 1- 3 X 2 + 2 is normal with mean E ( Y 1 ) = E ( X 1 )- 3 E ( X 2 ) + 2 = 2 and variance var( Y 1 ) = var( X 1- 3 X 2 + 2) = var( X 1 ) + 9 var( X 2 )- 6 cov( X 1 ,X 2 ) = 1 + 9- 0 = 10, and Y 2 = 2 X 1- X 2- 1 is normal with mean E ( Y 2 ) = 2 E ( X 1 )- E ( X 2 )- 1 =- 1 and variance var( Y 2 ) = var(2 X 1- X 2- 1) = 4 var( X 1 ) + var( X 2 )- 4 cov( X 1 ,X 2 ) = 4 + 1- 0 = 5. Since cov( Y 1 ,Y 2 ) = cov( X 1- 3 X 2 + 2 , 2 X 1- X 2- 1) = 2 var( X 1 )- 7 cov( X 1 ,X 2 )+3var( X 2 ) = 2- 0+3 = 5, we conclude that Y = ( Y 1 ,Y 2 ) is multivariate normal N ( , ) where = 2- 1 and = 10 5 5 5 . 2 . Let B = 1 0 1 0 2 0 so that Y = B X . By Theorem 3.1, Y is MVN with mean B = 1 0 1 0 2 0 3 4- 3 = 8 and covariance matrix B B = 1 0 1 0 2 0 2 1 3 1 4- 2 3- 2 8 1 0 0 2 1 0 = 16- 2- 2 16 . 3 . Let B = 1- 1 2 1- 2- 2 0 3 so that Y = B X . By Theorem 3.1, Y is MVN with mean B = 1- 1 2 1- 2- 2 0 3 = and covariance matrix...
View Full Document

Page1 / 4

solutions07 - Stat 351 Fall 2007 Assignment #7 Solutions 1...

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