please, help me to solve this questions and provide detailed solutions

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Problem 3.1 Let x : ($1, . . . ,In) N N(0,In) be a MVN random vector in R". (a) Let U 6 Rnx\" be an orthogonal matrix (UTU : UUT : In) and nd the distribution of UTaz. Let y : (3,11, . . . ,yn) ~ N(0,E) be a MVN random vector in R\". Let E : UAUT he the spectral decomposition of E. (b) Someone claims that the diagonal elements of A are nonnegative. Is that true? ((3) Let 2 : U Ty and nd the distribution of z. ((1) Someone claims that the component of 2 are independent. Is that true? (e) What is cov(zi, zj) fort" # j? Here, 2,; is the ith component of z : (21,. . . , 2\"). What is var(zi)? (f) Let a : (a1,...,an) 6 R\" be a xed (nonrandom) vector, and nd the distri- bution of aTz. (g) Assume that A\": > 0 for all 2'. (Here, Aii is the 72th diagonal entry of A.) Can you choose a from part (f) to make var(aTz) : 1? If so, specify one such a. (h) Let 11.1 E R" be the rst column of U. Find the joint distribution of (aTz, ufy) E R2. (Note that this is two-dimensional vector.) Problem 3.2 Let H E Rn\" be symmetric and idempotent, hence a projection matrix. Let x w N (0, In). (a) Let a > 0 be a positive number. Find the distribution of 052:, (b) Let u 2 H2: and 7) = (I H):z: and nd the joint distribution of (11,11). (c) Someone claims that u and 1; are independent. Is that true? (d) Let ,u E Im(H). Show that He : p. (e) Assume that 1 E Im(H) and nd the distribution of 1TH22. Here, 1 : (1, . . . ,1) E IR" is the vector of all ones