solutions09 - Stat 351 Fall 2007 Assignment #9 Solutions 1....

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Unformatted text preview: Stat 351 Fall 2007 Assignment #9 Solutions 1. (a) By Definition I, we see that X 1- ρX 2 is normally distributed with mean E ( X 1- ρX 2 ) = E ( X 1 )- ρE ( X 2 ) = 0 and variance var( X 1- ρX 2 ) = var( X 1 ) + ρ 2 var( X 2 )- 2 ρ cov( X 1 ,X 2 ) = 1 + ρ 2- 2 ρ 2 = 1- ρ 2 . That is, X 1- ρX 2 = Y where Y ∈ N (0 , 1- ρ 2 ). Hence, Y = p 1- ρ 2 Z where Z ∈ N (0 , 1). In other words, there exists a Z ∈ N (0 , 1) such that X 1- ρX 2 = p 1- ρ 2 Z. 1. (b) Since X = ( X 1 ,X 2 ) is MVN, and since Z = X 1 p 1- ρ 2- ρX 2 p 1- ρ 2 , we conclude that ( Z,X 2 ) is also a MVN. Hence, we know from Theorem V.7.1 that the components of a MVN are independent if and only if they are uncorrelated. We find cov( Z,X 2 ) = cov X 1 p 1- ρ 2- ρX 2 p 1- ρ 2 ,X 2 ! = 1 p 1- ρ 2 cov( X 1 ,X 2 )- ρ p 1- ρ 2 var( X 2 ) = ρ p 1- ρ 2- ρ p 1- ρ 2 = 0 which verifies that Z and X 2 are, in fact, independent. Exercise 4.2, page 127 : If φ ( t,u ) = exp { it- 2 t 2- u 2- tu } = exp { it- 1 2 (4 t 2 + 2 tu + 2 u 2 ) } , then we recognize this as the characteristic function of a normal random variable...
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This note was uploaded on 03/26/2012 for the course STAT 351 taught by Professor Michaelkozdron during the Fall '08 term at University of Regina.

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solutions09 - Stat 351 Fall 2007 Assignment #9 Solutions 1....

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