GATE 2021 ST – Question 54

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 2 marks · Numerical answer

A centered bivariate normal density is f(x,y)=exp[(3/2)xy−(25/32)x²−2y²]/π on ℝ². Find 8E(XY).

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Correct answer: 3 to 3

Explanation

Match the exponent to −½(x,y)Σ⁻¹(x,y)ᵀ. The precision matrix is [[25/16,−3/2],[−3/2,4]],with determinant 4. Its inverse has off-diagonal entry (3/2)/4=3/8. Since means are zero,E XY=Cov(X,Y)=3/8. Therefore **8E XY=3**.

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