tut6_10

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The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859a Tutorial 6 1. Suppose ε 1 and ε 2 are independent standard normal random variables. (a) Identify the distribution of Y = 5 ε 1 +2 ε 2 , specifying any relevant parameter values. (b) Identify the distribution of Z = ε 2 1 + ε 2 2 , specifying any relevant parameter values. 2. Suppose X is a 30 × 6 matrix consisting of linearly independent columns. (a) Show that A = X ( X T X ) - 1 X T is a symmetric, idempotent matrix. (b) Find the trace of A . (c) Suppose ε is a vector consisting of 30 independent standard normal random vari- ables. Show that if B = ε T and C = ε T ( I - A ) ε , then B and C are independent. (d) Identify the distributions of B , C and D = 4 B/C , specifying any relevant parameter values. 3. Let ε denote a vector of eight independent standard normal random variables, and let A = 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 00 0 . 00 0 . 25 0 . 25 - 0 . 25 - 0 . 25 - 0 . 00 - 0 . 00 0 . 00 0 . 00 0 . 25 0 . 25 - 0 . 25 - 0 . 25 - 0 . 00 - 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25
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Unformatted text preview: . 00 . 00-. 25-. 25-. 00-. 00 . 00 . 00 . 25 . 25-. 25-. 25-. 00-. 00 . 00 . 00 . 25 . 25 , and B = 1 8 aa T where a T = [1 1 1 1-1-1-1-1]. (a) Is A symmetric? (b) Compute A 2 . (c) Identify the respective distributions of Y 1 = ε T Aε , and Y 2 = ε T ( I-A ) ε . (d) Are Y 1 and Y 2 independent? Hence, or otherwise, identify the distribution of 3 Y 1 /Y 2 . (e) Compute E [ Y 1 ] and E [ Y 2 ]. (f) Is B idempotent? (g) Compute AB and determine ( A-B ) 2 . (h) Identify the respective distributions of W 1 = ε T Bε and W 2 = ε T ( A-B ) ε . (i) Are W 1 and W 2 independent? Hence, or otherwise, identify the distribution of W 1 /W 2 ....
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