exam02s - THE UNIVERSITY OF CHICAGO Graduate School of...

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THE UNIVERSITY OF CHICAGO Graduate School of Business Business 41912, Spring Quarter 2002, Mr. Ruey S. Tsay Solutions to Midterm 1. Make use of properities of multivariate normal distribution and that of matrix to answer this question. (a) What is the distribution of X = ( Z 1 , Z 2 ) 0 ? Answer: X N 2 ˆ" 3 2 # , " 2 1 1 2 #! . (b) What is the conditional distribution of Z 1 given that Z 3 = 3? Answer: Z 1 and Z 3 are independent so that Z 1 | Z 3 N (3 , 2). (c) Find a linear combination y = C 0 X such that C 0 C = 1 and Var( y ) is as large as possible. What is the value of the resulting Var( y )? Answer: The eigenvalues of the covariance matrix of X are 3 and 1. Speci±cally, let A be the covariance matrix of X . Solve the equation | A - λ I | = 0 to obtain the two eigenvalues. The eigenvectors are ( 1 2 , 1 2 ) 0 and ( 1 2 , - 1 2 ) 0 , respectively. Consequently, C = ( 1 2 , 1 2 ) 0 and Var( y ) = 3. (d) What are the eigenvalues of Σ ? Answer: From the block diagonal structure of Σ . The eigenvalues are 3, 1, 2. (e) Find the squared root matrix of Σ . Answer: Again, from the block diagonal structure of Σ , we can ±nd the squared root of the covariance matrix A of X ±rst, which is " 1 2 1 2 1 2 - 1 2 #" 3 0 0 1 #" 1
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exam02s - THE UNIVERSITY OF CHICAGO Graduate School of...

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