In: Statistics and Probability
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.