GATE 2021 ST – Question 45

Joint Distributions · Joint and Conditional Distributions · 2 marks · Multiple choice

X₁,X₂,X₃ are uncorrelated with common variance σ². Let Y₁=2X₁+X₂+X₃,Y₂=X₁+2X₂+X₃,Y₃=X₁+X₂+2X₃. P:the covariance-matrix eigenvalues of (Y₁,Y₂,Y₃) sum to 18σ². Q:Corr(Y₁,Y₂)=Corr(Y₂,Y₃). Which hold?

P only
Q only
Both P and Q
Neither P nor Q

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Show answer and explanation

Correct answer: (C) Both P and Q

Explanation

Each Y has variance (4+1+1)σ²=6σ². The eigenvalue sum equals the trace,18σ²,proving P. Every pair has covariance (2+2+1)σ²=5σ²,so all pairwise correlations equal 5/6,proving Q. **C**.

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