GATE 2025 ST – Question 49
Independent standard normal X₁,X₂,X₃ define Y₁=X₁−X₂,Y₂=X₁+X₂−2X₃,Y₃=X₁+X₂+X₃. Which hold?
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Correct answer: (A) Y₁,Y₂,Y₃ are independent.; (C) 2Y₃/√(3Y₁²+Y₂²)~t₂
Explanation
The coefficient rows are mutually orthogonal, so the jointly normal Y_i are independent with variances 2,6,3. Put Z₁=Y₁/√2,Z₂=Y₂/√6,Z₃=Y₃/√3, independent standard normals. C becomes Z₃/√[(Z₁²+Z₂²)/2], which is t₂. The unscaled squared sum in B is not χ²₃; D is F₂,₁, not F₁,₁. **A,C**.