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mid1 2007 - The University of Western Ontario Department of...

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The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 359a Quiz 1 DO ANY TWO OF THE THREE (EQUALLY WEIGHTED) QUESTIONS. Time Allowed: 60 Minutes Total Marks Available: 20 1. Write your answers in the spaces provided. Suppose P = 1 6 5 2 - 1 2 2 2 - 1 2 5 . and suppose ε 1 , ε 2 and ε 3 are independent standard normal random variables. (a) Identify the distribution of Y 1 = ε 1 + ε 2 + ε 3 , specifying any relevant parameter values. (b) Determine whether Y 1 is uncorrelated with Y 2 , where Y 2 = [1 - . 5 - . 5] ε = ε 1 - 1 2 ε 2 - 1 2 ε 3 . (c) Is Y 1 independent of Y 2 ? Explain briefly. (d) Identify the distribution of 2 Y 2 2 / 3, specifying any relevant parameter values. (e) Identify the distribution of Y 1 2 Y 2 2 , specifying any relevant parameter values. (f) Show that P is symmetric and idempotent, and find its trace. (g) Identify the distribution of ε T P ε , specifying any relevant parameter values.
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