1 (Conditional distribution of multivariate normal distribution). (40 pts). Consider a multivariate normal distribution. Exx Ery Erz x fx Y 2 ~N " yx 0 zx 0 zz where yy and Ezz are non-singular. (a) Let f(y) = x + xyyy (y- y), and define r = x - f(y). Calculate the mean vector and the covariance matrix of r. (b) Are r and y independent? Why? (c) Write x = f(y) +r. According to Part (a) and (b), what is the conditional mean of a given y? What is the conditional covariance matrix of a given y? (d) Now make use Part (b) to show that (2|y, z) = + (y - My) + (2 ) Var(x|y, z) = Exx - xyyy Lyx Exz Ezz Exx