Examination Paper for STAT0005 Page 3 Section A A1 Let X U(1, 1) and let Y = X4. (a) Compute E[Y ]. [3] (b) Compute Var(Y ). [3] (c) Compute the pdf fY of Y . [5] A2 Let the joint distribution of X and Y be given by the following two-way table: X Y -1 0 1 -1 b 0 a 0 0 1-2a-2b 0 1 a 0 b Here, a, b are unknown constants such that the above table is a valid two-way table. For parts (a)-(d) you may leave your results in terms of a and b where necessary. (a) Compute the marginal pmfs pX and pY . [3] (b) Compute E[X] and E[Y ]. [3] (c) Compute Var(X) and Var(Y ). [3] (d) Compute Corr(X, Y ) in the case (a, b) 6= (0, 0). [5] (e) What constraints must the pair (a, b) satisfy to ensure that the table above is valid [2] (f) What is the smallest value Corr(X, Y ) can take in this case Give a value of (a, b) for which the smallest possible value of Corr(X, Y ) is attained. [2] A3 Let the joint cdf of X and Y be given by FX,Y (x, y) = 8<: 0 if x < 0 or y < 0min{x, y} if x, y 0 and (x 1 or y 1) 1 if x, y 1 . (a) Compute P (0 < X 1, 0 < Y 1). [2] (b) Compute the marginal cdf FY of Y . [3] (c) Compute P (X < 1/2 |Y < 1/2). [Type] Using only words and no formulae, decide whether X and Y are independent and justify your decision. Maximum length: 150 words. [5] Turn Over Examination Paper for STAT0005 Page 4 Section B B1 For i 2 {1, . . . , n} with n 2 N and n 2, let Xi i.i.d. N(μ, 2) where μ and 2 > 0 are both unknown. To estimate the variance 2, consider the estimator T = 1 n nX i=1 (Xi Xˉ)2, where Xˉ = 1n Pn i=1Xi. In this expression, the number 2 (1, 2) is used to obtain di erent estimators. (a) Name the estimator in the case = 1. What is the expected value of T1 (You do not need to compute the expected value if you know it.) [2] (b) Compute the bias of T . [2] (c) Compute the sampling variance of T . Hint: Start from the variance of T1. [5] (d) Show that the mse for T is given by mse(T ; 2) = 4 (n )2 2 2 1 + 2n . [4] (e) Given the sample size n, which value (n) of results in the smallest mean square error of T You need to provide a derivation and justification of your result and while you may omit checking the second order condition you should check the boundaries. [8] (f) Hence provide a formula for an estimator of the form T with smallest mse. [Type] Using only words and no formulae, give one reason why T1 is often used in practice in spite of your result. Maximum length: 300 words. [5] Continued Examination Paper for STAT0005 Page 5 B2 For n 2 N, consider the regression model of the form Y = x+ , N(0, ), where the covariate x 2 Rn with x 6= 0 and the positive definite sym- metric matrix 2 Rn n are fixed and known whereas the parameter 2 R is unknown. The sample consists of a single observation y from this model. (a) [Type] Using only words and no formulae, expl
ain 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 given the one observation y 2 Rn. [4] (c) Show that the MLE of is given by b MLE = xT 1y xT 1x . You need to check all applicable conditions for a maximum. Hint: Note that is a number (not a matrix or a vector) and note the dimension of xT 1y as well as that of xT 1x. [8] (d) Compute E[b MLE]. [5] (e) Compute the Cramer-Rao lower bound for unbiased estimators of . What is the interpretation of this bound [4] (f) Compute the sampling variance of b MLE. Hence decide whether or not b MLE achieves the Cramer-Rao lower bound. Hint: First show that b MLE is of the form b MLE = c+ aT for some constant c 2 R and some constant vector a 2 Rn which you should specify. [8] Turn Over Examination Paper for STAT0005 Page 6 Section C Let X N(μ, ) where μ 2 Rn and 2 Rn n for dimension n 2. Also assume that is positive definite symmetric. Let A 2 Rk n with k 2 {1, . . . , n}. You may use the facts that, firstly, A AT 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(Xi, Xj) = 0 if and only if Xi and Xj 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, Xj) = 0 holds for all j 2 {2, . . . , d}. Show that X1 is independent of X2 +X3 + . . .+Xd. [5] (e) Consider three discrete random variables U, V,W each taking values in {1, 1} and such that U is independent of V and U is independent of W . For each of the statements (i) and (ii) below, decide whether it is true in general or whether it may be false. If the statement is true in general, provide a proof. Otherwise, find a joint pmf for U, V and W that provides a counterexample (i.e. a joint distribution for which the statement does not hold), explaining your reasoning and presenting your joint pmf in a table as follows: u v w pUVW (u, v, w) -1 -1 -1 . . . 1 -1 -1 . . . … … … (i) U and V +W are uncorrelated. [4] (ii) U and V +W are independent. [5] (f) [Type] Using only words and no formulae, write a short essay on three ways in which the Gaussian distribution is special among probability distributions. You need to make clear how the ways are directly related to this Section C and/or to results in the lecture notes. Maximum length: 300 words. [10] End of Paper