B2 For n E N, consider the regression model of the form Y = arte, E~ N(0, E), where the covariate r E R" with x 0 and the positive definite sym- metric matrix E e R"x" are fixed and known whereas the parameter o E R is unknown. The sample consists of a single observation y from this model. (a) [Type] Using only words and no formulae, explain how a maxi- mum likelihood estimator is obtained in general. Maximum length: 150 words. [4] (b) In the setting described above, obtain the log-likelihood for the parameter o given the one observation y e R". [4] (c) Show that the MLE of a is given by OMLE = xE-ly You need to check all applicable conditions for a maximum. Hint: Note that a is a number (not a matrix or a vector) and note the dimension of a? E-ly as well as that of a? E-lx. 8 (d) Compute E[aMLE]. [5]Section C Let X ~ N(u, E) where u E R" and E E Rx for dimension n > 2. Also assume that E is positive definite symmetric. Let A e R*xn with ke {1, ...,n}. You may use the facts that, firstly, AZA is positive definite symmetric if A has full rank and that, secondly, A has full rank if and only if its row vectors are linearly independent. (a) Compute the mgf of Y = AX and thus show that Y follows a normal distribution and specify its mean vector and covariance matrix. [4] (b) Under the condition that A has full rank, write down the pdf of Y. Explain why it is impossible to write down the pdf of Y if A does not have full rank. [4] (c) Using the mgf or otherwise, prove that Cov(X,, X;) = 0 if and only if X; and X, are independent. Note that during the course we have only established this in the case of bivariate normal distributions. [6] (d) Suppose that Cov(X1, X;) = 0 holds for all je {2, ..., d}. Show that X1 is independent of X2 + X3 + ... + Xd. [5]