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Question 1 Suppose that X1 s N(1; 2); X2 s N(4; 3) and Cov(X1; X2) = 1

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Question 1 Suppose that X1 s N(1; 2); X2 s N(4; 3) and Cov(X1; X2) = 1. Let X = X1 X2 ; A = 1 1 2 1 ; Y = Y1 Y2 and Y = AX: In a matrix form Önd (a) the mean of Y: (b) the variance-covariance matrix of Y: Show your working. Question 2 A researcher in the educational Öeld has collected the following test scores from a group of six students (n = 6). Student Language Aptitude x1 Analogical Reasoning x2 Geometric Reasoning x3 1 2 3 15 2 6 8 9 3 5 2 7 4 9 4 3 5 11 10 2 6 1 4 12 Let X = [x1 ; x2; x3]; x = [x1; x2; x3] T and 1 = [1; 1; 1; 1; 1; 1]T be the data matrix, sample mean column vector and all-ones column vector, respectively. (a) Find x: (b) Calculate 1x T : Show your working. (c) Calculate X 1x T . Show your working. (d) Calculate the SSCP (sum of squares and cross-products) matrix. Show your working. (e) Find S, the sample covariance matrix. (f) Is S a positive deÖnite matrix? Provide your reasoning. (g) Hence using S, Önd R, the sample correlation matrix. Question 3 Use the following R-code and obtain matrix C: rm(list=ls()) A <- matrix(rpois(64,5), 8, 8) A C= A %*% t(A) C (a) Provide a detailed description of matrix A including the dimension and commenting on how its elements were obtained. Hint: Use R-help: ?rpois( ) . (b) Obtain all eigen values and eigenvectors of matrix C. Provide the R-code and output. (c) Is C a positive deÖnite matrix? Explain your answer. (d) Let the matrix of the eigenvectors be U. Verify U TU = I using R. (e) Verify the SVD (singular value decomposition) for C using R.