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a2_09 - The University of Western Ontario Department of...

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The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859A – Assignment 2 Due Date: 23 October, 2009 1. Suppose { Y i } is a sequence of n independent normal random variables with mean μ and variance σ 2 . Denote the column vector consisting of these variables as Y , let 1 denote the column vector consisting of n 1’s, and let J = 11 T . (a) Show that 1 T Y is uncorrelated with each component of ( I - J/n ) Y . Deduce stochas- tic independence in each case. (b) Show that 1 T Y is stochastically independent of Y T ( I - J/n ) Y . Deduce that ¯ Y (the sample mean) is stochastically independent of S 2 , the sample variance of the Y i ’s. (c) Show that ¯ Y - μ S/ n has a t -distribution on n - 1 degrees of freedom. 2. Use R for this problem. Look up the help information on the functions det , t , eigen and solve . Write a sentence describing what each of these functions is used for. 3. Use R for this problem. Assign the following matrix to an object called harmonic.mat . A = 1 1 / 6 1 / 11 1 / 16 1 / 21 1 / 2 1 / 7 1 / 12 1 / 17 1 / 22 1 / 3 1 / 8 1 / 13 1 / 18 1 / 23 1 / 4 1 / 9 1 / 14 1 / 19 1 / 24 1 / 5 1 / 10 1 / 15 1 / 20 1 / 25 (a) Obtain the transpose of A , and compute the product A T A .
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