Question

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(0, \Sigma)\), where \(\Sigma\) is a positive definite matrix. Then which of the following statements is/are correct?

• A. \(X_1, \ldots, X_p\) are always independent normal random variables
• B. \(X_1, \ldots, X_p\) are normal random variables
• C. \(X_1^2 + \cdots + X_p^2\) always follows a chi-square distribution
• D. Any linear combination of \(X_1, \ldots, X_p\) is a normal random variable

Hint:

Marginals and linear combinations of a multivariate normal vector are always normal, but independence needs a diagonal \(\Sigma\), and \(\sum X_i^2\) is chi-square only when \(\Sigma = I\).

Answer 1

The Correct Option is B, D

Step 1: Defining property.
Every linear combination of MVN vector is normal.
Step 2: (D) holds directly.
TRUE.
Step 3: (B) is special case.
Each \(X_i=e_i^TX\) normal. TRUE.
Step 4: (A) fails.
Independence needs diagonal \(\Sigma\), not given. FALSE.
Step 5: (C) counterexample.
\(\Sigma=\text{diag}(4,1)\) breaks chi-square shape. FALSE.
\[ \boxed{\text{(B) and (D)}} \]