Question 1. (20 Marks) Let X=
(a) Find the best linear approximation of X3 by a linear function of X1 and X2. (5 Marks)
(b) Using R, simulate n = 1000 samples from X. Transform the data to Z = ( 21 22 23)
where Z1 = X2 – X3, Z2 = X2+ Xy and Z3 = Z1 + Z2. Plot the scatter-plots of pairs of
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observations and do the test to confirm the multivariate normality of Z. To generate the data,
set the seed equal to the last 3 digits of your student zID; i.e., if your student zID is 1234567,
you need to use “‘set .seed (567)”. Interpret the result of the test. (15 Marks)
Question 2. (25 Marks)
(a) State, explicitly, all possible values that a and b can take in order for the following matrix to
be a covariance matrix. Give arguments that justify your answer 5 Marks
(b) Withou using using R, compute the eigenvalues and the eigenvectors of the matrix 10 Marks
(c) Using R, confirm the eigenvalues and the eigenvectors of E obtained in (b) and define the
matrix P and the diagonal matrix A such that y = PApT. (10 Marks)
Question 3. (25 Marks) Consider X
(a) Show that a7 X and b” X are independent. [5 Marks]
(b) Using R, simulate n = 1000 samples from X and use the appropriate plot to confirm inde-
pendence of a7X and b7 X. visually. Make sure to set the seed equal to the last 3 digits of
your student zID, as in Ouestion 1. [10 Marks)
(c) Assume that the mean and covariance matrix of the simulated data is unknown. Find the
ML estimators for u and E based on the sample. [10 Marks)
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