Question

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(\mu, \Sigma)\), where \(\mu \in \mathbb{R}^p\) and \(\Sigma\) is a positive definite matrix. Let \(A\) be a \(p \times p\) matrix such that \(A^T A = I_p\).
Let \(Y = (Y_1, \ldots, Y_p)^T = AX\). Then which of the following statements is/are correct?

• A. \(Y\) follows \(N_p(A\mu, A^T \Sigma A)\)
• B. \(Y\) follows \(N_p(A\mu, A \Sigma A^T)\)
• C. \(\text{Var}\left(\sum_{i=1}^{p} Y_i\right) = \text{Var}\left(\sum_{i=1}^{p} X_i\right)\)
• D. \(\sum_{i=1}^{p} \text{Var}(Y_i) = \sum_{i=1}^{p} \text{Var}(X_i)\)

Hint:

Use \(\text{Cov}(AX) = A\Sigma A^T\) and the cyclic property of trace, \(\text{tr}(A\Sigma A^T) = \text{tr}(\Sigma A^TA) = \text{tr}(\Sigma)\), remembering \(A^TA = I_p\).

Answer 1

The Correct Option is B, D

Step 1: A is orthogonal.
\(A^TA=I_p\Rightarrow AA^T=I_p\).
Step 2: Formula.
\(E(Y)=A\mu\), \(\text{Cov}(Y)=A\Sigma A^T\). (A) FALSE, (B) TRUE.
Step 3: (C).
\(\text{Var}(\sum Y_i)=(A^T\mathbf{1})^T\Sigma(A^T\mathbf{1})\), not always equal to \(\mathbf{1}^T\Sigma\mathbf{1}\) unless \(A^T\mathbf{1}=\mathbf{1}\). FALSE in general.
Step 4: (D).
\(\sum\text{Var}(Y_i)=\text{tr}(A\Sigma A^T)=\text{tr}(\Sigma A^TA)=\text{tr}(\Sigma)=\sum\text{Var}(X_i)\). TRUE.
Final Answer: \[ \boxed{\text{(B) and (D)}} \]