GATE 2021 ST – Question 19
Independent vectors X₁,X₂,X₃∼N₄(0,Σ₁) and Y₁,…,Y₄∼N₄(0,Σ₂) have positive definite covariance matrices. Let X=[X₁ X₂ X₃],Y=[Y₁ Y₂ Y₃ Y₄],and Z=Σ₁^(−1/2)XXᵀΣ₁^(−1/2)+Σ₂^(−1/2)YYᵀΣ₂^(−1/2). Which law does Z follow? W_m(ν,Σ) denotes Wishart order m,df ν,scale Σ.
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Correct answer: (A) W₄(7,I₄)
Explanation
Whitening each vector gives independent standard four-dimensional normals. Their outer-product sums are Wishart₄(3,I₄) and Wishart₄(4,I₄). Independence and the common scale make their sum Wishart₄(7,I₄). **A**.