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?
Show Hint
- \(X_1, \ldots, X_p\) are always independent normal random variables
- \(X_1, \ldots, X_p\) are normal random variables
- \(X_1^2 + \cdots + X_p^2\) always follows a chi-square distribution
- Any linear combination of \(X_1, \ldots, X_p\) is a normal random variable
The Correct Option is B, D
Solution and Explanation
Independence needs \(\Sigma\) diagonal; not given. FALSE.
Step 2: (B).
Marginals of multivariate normal always normal. TRUE.
Step 3: (C).
Only chi-square if \(\Sigma=I\); general \(\Sigma\) fails. FALSE.
Step 4: (D).
Defining property of MVN: any linear combination is normal. TRUE.
Final Answer: \[ \boxed{\text{(B) and (D)}} \]
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