GATE 2025 ST – Question 45

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

Five iid bivariate N₂(0,I₂) vectors form rows of a 5×2 matrix X. Let W=XᵀX and Z=W₁₁+4W₁₂+4W₂₂. Find Var Z.

Show answer and explanation

Correct answer: (C) 250

Explanation

Z is the sum over five rows of (X_i1+2X_i2)². Each linear form is independent N(0,5), so Z/5~χ²₅. Therefore Var Z=25×(2·5)=**250, C**.

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