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GATE 2020 ST – Question 48

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

X∼N₃((1,2,3),Σ), where $$\Sigma=\begin{pmatrix}1&.5&0\\.5&1&0\\0&0&5\end{pmatrix}.$$ Which pair is independent?

  1. X₁−X₂+X₃ and X₁
  2. X₁+X₂ and X₃−X₁
  3. X₁−X₂+X₃ and X₁+X₂
  4. X₁−2X₂ and2X₁+X₂

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Show answer and explanation

Correct answer: (C) X₁−X₂+X₃ and X₁+X₂

Explanation

For jointly normal linear combinations, zero covariance implies independence. In C, expanding the covariance gives Var(X₁) − Var(X₂) + Cov(X₃,X₁) + Cov(X₃,X₂). The X₁/X₂ cross terms cancel. The remaining value is 1 − 1 + 0 + 0 = 0. The other proposed pairs have nonzero covariances. Therefore C is the independent pair.

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