GATE 2024 ST – Question 33

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 1 mark · Numerical answer

A centered bivariate normal (X,Y) satisfies E(Y|X=2)/E(X|Y=1)=8. Find Var(Y|X=2)/Var(X|Y=1) (integer).

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Correct answer: 4 to 4

Explanation

Write marginal standard deviations σ_X,σ_Y and nonzero correlation ρ. The conditional-mean ratio is [2ρσ_Y/σ_X]/[ρσ_X/σ_Y]=2σ_Y²/σ_X²=8. Thus σ_Y²/σ_X²=4. Both conditional variances contain the same factor 1−ρ², so their ratio is **4** under the nondegenerate conditional-variance convention.

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