Question

Problem 4 (20 pts): Let $X \sim N_p(\mu, I_p)$ and $A_i$ be an $p \times p$ symmetric matrix satisfying $A_i^2 = A_i$, $i = 1, 2$. Show that $X^T A_1 X$ and $X^T A_2 X$ are independent if and only if $A_1 A_2 = 0$.

          Problem 4 (20 pts): Let $X \sim N_p(\mu, I_p)$ and $A_i$ be an $p \times p$ symmetric matrix satisfying

$A_i^2 = A_i$, $i = 1, 2$.

Show that $X^T A_1 X$ and $X^T A_2 X$ are independent if and only if $A_1 A_2 = 0$.

Problem 4 (20 pts): Let X ∼ Np(μ, Ip) and Ai be an p × p symmetric matrix satisfying

Ai^2 = Ai, i = 1, 2.

Show that X^T A1 X and X^T A2 X are independent if and only if A1 A2 = 0.

Added by Patricia G.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

11/22/2023

Step 1

Since X ~ Np(μ,Σ), we can express X as X = μ + AZ, where Z ~ Np(0, I) and A is the Cholesky decomposition of Σ, such that Σ = AA^T. Show more…

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Problem 4 (20 pts): Let X ~ Np(μ,Σ) and A be a p x p symmetric matrix satisfying A^T = A, i=1,2. Show that X^T AX and X^T A2X are independent if and only if AΣA2 = 0.