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GATE 2021 ST – Question 52

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

(X₁,X₂,X₃,X₄)∼N₄(μ,Σ),μ=(1,0,0,1)ᵀ and Σ has rows (1,.2,0,0),(.2,2,0,0),(0,0,2,.2),(0,0,.2,1). Which holds?

  1. (5/17)[(X₁+X₂)²+(X₃+X₄−1)²]∼central χ²₂.
  2. [(X₁+X₃−1)²+(X₂+X₄−1)²]/3∼central χ²₂.
  3. E√|(X₁+X₂−1)/(X₃+X₄−1)| is not finite.
  4. E|(X₁+X₂+X₃+X₄−2)/(X₁+X₂−X₃−X₄)| is not finite.

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Correct answer: (D) E|(X₁+X₂+X₃+X₄−2)/(X₁+X₂−X₃−X₄)| is not finite.

Explanation

Let A=X₁+X₂−1 and B=X₃+X₄−1. They are independent centered normals of equal variance 17/5. The ratio (A+B)/(A−B) is standard Cauchy,because its numerator and denominator are independent equal-variance centered normals. Its absolute first moment diverges,proving **D**. A has an uncentered first term;B’s two linear forms have nonzero covariance .4;C’s square-root absolute Cauchy moment is finite since its order is below one.

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