Suppose the random variables X1 and X 2 have the covariance matrix 2 1 E - ( 1 4 l with the eigenvalues and corresponding eigenvectors given as follows A1 = 4414, el = (0.383,0.924)T, A2 2 1.586, .22 = (0.924, 0.383)T. (a) [1 marks] Use the above eigenvalues and eigenvectors, write down the singular value decomposition expression for E. (b) [3 marks] Explain how to generate random vectors from the multivariate normal distribution N01, 2), where a = (050)1' and E is given above. (c) [2 marks] Use the following four N (0, 1) random numbers 0.626, 0.184, 0.836, 1.5 to generate 2 random vectors (each has the dimension of 2 X 1) from a N(u, E) with it : (0,0)T. ((1) [2 marks] Calculate the factor loadings if we use only the second eigen value and vector (i.e. 62 and A2 ) to perform Factor Analysis. (c) [2 marks] Hence calculate the corresponding specic variances for the two random variables. (f) [2 marks] Using your answers in (d) and (e) express the random vari ables X1, X2 in terms of the obtained factor loadings