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Let X1 and X2 be two i.i.d. p × 1 multivariate normal random vectors with mean μ and positive definite dispersion matrix ∑. Then which of the following random variables always has a central chi-square distribution 

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. \(\frac{1}{2}\)(X1 - X2)T (X1 - X2)
  2. 2(X1 - X2)T (X1 - X2)
  3. 2(X1 - X2)T ∑-1(X1 - X2)
  4. \(\frac{1}{2}\)(X1 - X2)T ∑-1(X1 - X2)​

Answer (Detailed Solution Below)

Option 4 : \(\frac{1}{2}\)(X1 - X2)T ∑-1(X1 - X2)​
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Detailed Solution

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\(\frac12\)(X1 - X2)T ∑-1(X1 - X2)​ has a central chi-square distribution 

Option (4) is correct

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Last updated on Jul 8, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website.The exam will be held on 28th July 2025.

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