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

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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 = 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 = 1 - ρ 2 Z. 1. (b) Since X = ( X 1 , X 2 ) is MVN, and since Z = X 1 1 - ρ 2 - ρX 2 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 1 - ρ 2 - ρX 2 1 - ρ 2 , X 2 = 1 1 - ρ 2 cov( X 1 , X 2 ) - ρ 1 - ρ 2 var( X 2 ) = ρ 1 - ρ 2 - ρ 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 X = ( X 1 , X 2 ) N 1 0 , 4 1 1 2 .
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