GATE 2023 ST – Question 41

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

X₁,…,X₁₀ are iid N₂(μ,Σ),Σ nonsingular. With X̄ their mean and U=1/[1+(X̄−μ)ᵀΣ⁻¹(X̄−μ)],find ln P(U≤1/2).

Show answer and explanation

Correct answer: (A) −5

Explanation

The statistic Q=10(X̄−μ)ᵀΣ⁻¹(X̄−μ) is χ²₂. U≤1/2 is equivalent to Q≥10. A χ²₂ variable has survival exp(−q/2),so P=e⁻⁵ and its log is **−5, A**.

GATE 2022 ST Q38 – Four independent observations (3,6)ᵀ,(4,4)ᵀ,(5,7)ᵀ,(4,7)ᵀ come from a bivariate normal…
GATE 2022 ST Q65 – X=(X₁,X₂,X₃)ᵀ∼N₃(μ,Σ),μ=(3,2,4)ᵀ,and Σ has rows (4,2,1),(2,3,0),(1,0,2). Find E(X₁…
GATE 2021 ST Q52 – (X₁,X₂,X₃,X₄)∼N₄(μ,Σ),μ=(1,0,0,1)ᵀ and Σ has rows…
GATE 2021 ST Q54 – A centered bivariate normal density is f(x,y)=exp[(3/2)xy−(25/32)x²−2y²]/π on ℝ². Find…
GATE 2026 ST Q35 – (X₁,X₂,X₃)ᵀ is normal with mean (1,1,1)ᵀ and covariance…
GATE 2026 ST Q47 – Independent iid X,Y follow N p(μ,Σ),Σ positive definite. Which follows a chi-square law as…