Let a: = (1191,\" . ,mn) ~ N(0,In) be a MVN random vector in R\". (a) Let U 6 RM\" be an orthogonal matrix (UTU = UUT = n) and nd the distribution of UTm. Let y = (311,. . . ,yn) ~ NH], 2) be a MVN random vector in R\". Let 2 = UAUT be the spectral decomposition of 2. (b) Someone claims that the diagonal elements of A are nonnegative. Is that true? (c) Let 2 = UTy and nd the distribution of z. (d) Someone claims that the component of z are independent. Is that true? ) (e What is cov(zz-, zj) for t' a 3'? Here, zz- is the ith component of z = (zl, . . . ,2\"). What is var(zz-)? (f) Let a, = (0.1, . . . ,an) E R" be a xed (nonrandom) vector, and nd the distri- bution of aTz. (g) Assume that A\"- > 0 for all 3'. (Here, A\"- is the ith diagonal entry of A.) Can you choose a from part (f) to make var(aTz) = 1? If so, specify one such a. (h) Let in E R\" be the rst column of U. Find the joint distribution of (aTz, ugly) E R2. (Note that this is two-dimensional vector.)