Simulate 10000 values of y - microsum-y - micro where sum-

Question

Q1. Let x ∼ Nk (0, σ²I) and let y = px, where P is a k x k orthogonal matrix. Show that y has a N_k (0, σ²I) distribution.

Q2. Suppose Y ∼ N₃ (0, I₃).

(a) Show that Q = 1/3[(Y_l - Y₂)² + (Y₂ - Y₃)²  + (Y₃ - Y₁)²] has a χ²(2) distribution.

(b) Find the distribution of. What can you tell about the density of V?

Q3. Let Z ∼ N₃(0, I₃)and Y = µ + AZ, where

(a) Use R to find the mean and variance of Y_max = max{Y₁, Y₂, Y₃, Y₄} by simulation.

(b) Simulate 10000 values of (Y - µ)'∑^-(Y - µ), where ∑^- is a g-inverse for the covariance of Y. Find the mean and variance of the simulated values. Also draw the histogram and the density of the simulated values.