GATE 2026 ST – Question 35

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 1 mark · Numerical answer

(X₁,X₂,X₃)ᵀ is normal with mean (1,1,1)ᵀ and covariance $\Sigma=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}$. Find Var(X₁|X₂=1,X₃=−1) to two decimals.

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Correct answer: 1.30 to 1.36

Explanation

The conditional covariance formula gives 2−(1,1)[[2,1],[1,2]]⁻¹(1,1)ᵀ. The inverse is (1/3)[[2,−1],[−1,2]], so the subtracted term is 2/3. Conditional variance=4/3=**1.33**, independent of the observed conditioning values.

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