Question 2. [26 marks] (a) [6 marks] Consider a p-dimensional random vector X where X-Np(0px1, [pxp). Answer the follow questions. (i) What is the distribution of X1 - 2X2, where X] and X2 are both from the above normal distribution? (You need to specify the distribution, its mean and covariance matrix.) (ii) Briefly explain why XTE-1X follows x (p) distribution. (iii) Suppose App is a symmetric and positive definite matrix and therefore there exists a "square-root" matrix A1/2 such that A = A1/ZA1/2 . Find the distribution of A1/ZX. (b) [5 marks] Let X1. ... Xn be a multivariate random sample, where each X, has the dimension of px1. (i) Explain how to check if a random sample X]. .... X, follow a multivariate normal distribution. (ii) If X1. .. Xn fail the normality checking, discuss how to transform each component of these random vectors to univariate normal random variables. (c) [4 marks] Consider two random variables (X1, X2) (i) If X] follows a normal distribution and X2 is also normally distributed, will (X1,X2) jointly follow a bi-variate normal distribution? (No need to give a reason.) (ii) If (X1.*2) follow a bi-variate normal distribution, will X, or X2 are also normally distributed? Why? (In this part, you need to provide reasons briefly when answering "why" .) (d) [5 marks] Suppose 15 multivariate observations are obtained on 3 variables and these observations are denoted by X1. ....*js, where each x, is a 3 x 1 vector containing 3 values from the 3 variables. Consider the population mean vector # = (#,. /2, #;). Suppose we wish to test the following hypotheses involving different contrasts: Ho: M1- 2 Mi + 3 = 0.P3 2 P1 # #2 - 0, 2 and significance level for the test is a = 5%. Clearly explain how to test this hypothesis. In your answer you must include: (i] assumptions made, (ii) test statistic, and (iii) how to decide if to reject or retain the null hypothesis. (e) [6 marks] This question considers the data and contrasts specified in part (d) above. Simultaneous confidence intervals are often needed, particularly when the null hypothesis is rejected. Answer the following questions (with the overall confidence level of 95%%). (i) Write down the T' simultaneous confidence intervals for all the contrasts. You are required to simplify the answers to as much as you can. (ii) Write down the Bonferroni simultaneous confidence intervals for all the contrasts. You are required to simplify the answers to as much as you can