Suppose that X = (X1, ..., Xp) ~ Np (Mpx1, Epxp) is a random vector and Amxp is a non-random matrix. (a) [3 marks] What is the distribution of Y = AX? (Specify the distribution, its mean and covariance matrix. The derivation is not required.) (b) [3 marks] What is the distribution of X1, the first element of X? Show how to obtain this result from (a). (c) [4 marks] Let Apx1 = (0, ...,0) . Explain why in this case X E-1X follows x'(p) distribution. (d) [3 marks] Two random variables Z1 and Z2 are not independent. If Z1 follows a normal distribution and Z2 is also normally distributed, will (Z1, Z2) jointly follow a bivariate normal distribution? (Explain, no need to prove mathematically). (e) [2 marks] Let two random variables Z1 and Z2 be independent. If Z1 follows a nor- mal distribution and Z2 is also normally distribute then what is the distribution of (Z1, Z2) T? (f) [4 marks] Let Z = (Z1, Z2) T ~ N2(0, 12) be a bivariate standard normal vector, with 0 = (0,0) and 12 is 2 x 2 identity matrix. Suppose that Y = (Y1, Y2) = BZ, where N B = NO Determine the distribution of Y, including its parameter(s). Are Y, and Y2 indepen dent? Explain