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 rotates.
\(A^TA=I_p\) means \(A\) is orthogonal, a rotation/reflection.
Step 2: Y distribution.
\(Y\sim N_p(A\mu,A\Sigma A^T)\), matching (B) not (A).
Step 3: Projections argument.
\(\sum Y_i\) is projection onto \(A^T\mathbf{1}\), not necessarily same as \(\mathbf{1}\), so (C) fails generally.
Step 4: Trace invariance.
\(\sum\text{Var}(Y_i)=\text{tr}(A\Sigma A^T)=\text{tr}(\Sigma)\), rotation invariant. (D) TRUE.
\[ \boxed{\text{(B) and (D)}} \]