Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U...

Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U ∈ Rn×n be an orthogonal matrix (UTU = UUT = In) and nd the distribution of UTx. Let y = (y1,...,yn) ∼ N(0,Σ) be a MVN random vector in Rn. Let Σ = UΛUT be the spectral decomposition of Σ.
(b) Someone claims that the diagonal elements of Λ are nonnegative. Is that true?
(c) Let z = UTy and nd 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) ∈Rn be a xed (nonrandom) vector, and nd the distribution of aTz.
(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(aTz) = 1? If so, specify one such a. (h) Let u1,u2 ∈Rn be the rst and second columns of U. Find the joint distribution of 2uT 2 y−uT 1 y uT 1 y ∈R2. Note that this is a two-dimensional vector.

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(a)

(b)

orchestra answered 1 year ago

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