N(µ, Σ) denote a multivariate normal distribution (MVN). Problem 3.1 Let x = (x1, . . . , xn) ∼ N(0, In) be a MVN random vector in R n . (a) Let U ∈ R n×n be an orthogonal matrix (U TU = UUT = In) and find the distribution of U T x. Let y = (y1, . . . , yn) ∼ N(0, Σ) be a MVN random vector in R n . Let Σ = UΛU T be the spectral decomposition of Σ. (b) Someone claims that the diagonal elements of Λ are nonnegative. Is that true? (c) Let z = U T y and find the distribution of z. (d) What is the cov(zi , zj ) for i 6= j? Here, zi is the ith component of z = (z1, . . . , zn). What is the var(zi)? (e) Are the components of z independent? (f) Let a = (a1, . . . , an) ∈ R n be a fixed (nonrandom) vector, and find the distribution of a T z. (g) Assume that Λii > 0 for all i. (Here, Λii is the ith diagonal entry of Λ.) Can you choose a from part (f) to make var(a T z) = 1? If so, specify one such a. (h) Let u1, u2 ∈ R n be the first and second columns of U. Find the joint distribution of 2u T 2 y − u T 1 y u T 1 y ∈ R 2 . Note that this is a two-dimensional vector.
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