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GATE 2024 ST – Question 58

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 2 marks · Multiple select

(X₁,X₂,X₃)~N₃((2,−3,1)ᵀ,$\Sigma=\begin{pmatrix}25&-2&4\\-2&4&1\\4&1&9\end{pmatrix}$). For which a are X₂ and X₂−aᵀ(X₁,X₂)ᵀ independent?

  1. a=(0,1)ᵀ
  2. a=(−1,−1)ᵀ
  3. a=(0,2)ᵀ
  4. a=(2,2)ᵀ

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Correct answer: (A) a=(0,1)ᵀ; (D) a=(2,2)ᵀ

Explanation

The two variables are jointly normal, so zero covariance suffices (including a degenerate residual). Cov=4(1−a₂)−a₁Cov(X₂,X₁)=4(1−a₂)+2a₁. It vanishes for (0,1) and (2,2), and not for the other choices. **A,D**.

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