Question: Please help with this problem. Thank you! Problem 3.1 Let x = (x1, . .., In) ~N(0, In) be a MVN random vector in R.

Please help with this problem. Thank you!

Problem 3.1 Let x = (x1, . .., In) ~N(0, In) be a MVN random vector in R". (a) Let U E Rnxn be an orthogonal matrix (U'U = UU? = In) and find the distribution of UTx. Let y = (y1, . . . , Un) ~ N(0, E) be a MVN random vector in R". Let E = UAU' be the spectral decomposition of E. (b) Someone claims that the diagonal elements of A are nonnegative. Is that true? (c) Let z = Uly and find the distribution of z. (d) What is the cov(zi, 2;) for i * j? Here, zi is the ith component of z = (Z1, . . ., Zn). What is the var(zi)

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