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tut2_09

# tut2_09 - 00 1 00 1 00 0 00 0 00 0 00 00 0 00 2 00 0 00 0...

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The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859a Tutorial 2 1. Suppose ε 1 and ε 2 are independent standard normal random variables. (a) Identify the distribution of Y = 3 ε 1 - 4 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 15 × 10 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) List all 15 eigenvalues of A . (d) Suppose ε is a vector consisting of 15 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. (e) Identify the distributions of B , C and D = . 5 B/C , specifying any relevant param- eter values. 3. Suppose x and y are column vectors of length n . Show, from ﬁrst principles, that tr ( x T y ) = tr ( yx T ), where tr ( . ) denotes the trace operator. 4. Let ε denote a vector of ﬁve independent standard normal random variables, and let A = 1 2 1 . 00 1 . 00 0 . 00 0 . 00 0
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Unformatted text preview: . 00 1 . 00 1 . 00 0 . 00 0 . 00 0 . 00 . 00 0 . 00 2 . 00 0 . 00 0 . 00 . 00 0 . 00 0 . 00 1 . 00 1 . 00 . 00 0 . 00 0 . 00 1 . 00 1 . 00 , and B = 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . (a) Is A symmetric? (b) Compute A 2 . (c) What are the eigenvalues of A ? (d) Identify the respective distributions of Y 1 = ε T Aε , and Y 2 = ε T ( I-A ) ε . (e) Are Y 1 and Y 2 independent? Hence, or otherwise, identify the distribution of 2 Y 1 / (3 Y 2 ). (f) Compute E [ Y 1 ] and E [ Y 2 ]. (g) Compute B 2 , and identify the eigenvalues of B . (h) Compute AB , determine ( A-B ) 2 , and identify an eigenvector of A . (i) Identify the respective distributions of W 1 = ε T Bε and W 2 = ε T ( A-B ) ε . (j) Are W 1 and W 2 independent? Hence, or otherwise, identify the distribution of 2 W 1 /W 2 ....
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