GATE Multivariate Analysis: Multivariate Normal and Wishart Distributions – Previous Year Questions

19 GATE previous year questions on Multivariate Normal and Wishart Distributions (Multivariate Analysis, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q52 (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?
2. GATE 2021 ST Q54 (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).
3. GATE 2022 ST Q38 (2 marks, Multiple choice) – Four independent observations (3,6)ᵀ,(4,4)ᵀ,(5,7)ᵀ,(4,7)ᵀ come from a bivariate normal population. If μ̂,Σ̂ are their MLEs,find Σ̂μ̂.
4. GATE 2022 ST Q65 (2 marks, Numerical answer) – 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 X₂=4,X₃=7),as an integer.
5. GATE 2023 ST Q25 (1 mark, Multiple choice) – X₁,…,X n,Y₁,…,Y n are iid N p(μ,Σ),Σ nonsingular,p>1,n>1. Write their means X̄,Ȳ. Which claim is true?
6. GATE 2023 ST Q35 (1 mark, Numerical answer) – (X₁,X₂,X₃)∼N₃(0,Σ),where Σ has rows (2,2,1),(2,5,1),(1,1,1). Given Φ(−0.5)=0.3085,find P[(X₁−2X₂+2X₃)²<7/2],to two decimals.
7. GATE 2023 ST Q41 (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).
8. GATE 2024 ST Q33 (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).
9. GATE 2024 ST Q45 (2 marks, Multiple choice) – For 25 iid N₃(μ,Σ) observations,Σ nonsingular, let S=Σⱼ₌₁²⁵(X j−X̄)(X j−X̄)ᵀ/24 and B=pmatrix1&2&3\\-1&0&1pmatrix. Which describes 24BSBᵀ?
10. GATE 2024 ST Q58 (2 marks, Multiple select) – (X₁,X₂,X₃)~N₃((2,−3,1)ᵀ,=pmatrix25&-2&4\\-2&4&1\\4&1&9pmatrix). For which a are X₂ and X₂−aᵀ(X₁,X₂)ᵀ independent?
11. GATE 2025 ST Q23 (1 mark, Multiple choice) – A bivariate normal (X,Y) has means 3,4, variances 25,100 and covariance 50ρ,−1<ρ<1. If E(Y X=5)=4.32, find ρ.
12. GATE 2025 ST Q30 (1 mark, Numerical answer) – X is 10-dimensional N₁₀(0,I). Let Y=ln√(XᵀX) and M Y(t)=E(e(tY)),t>−10. Find M Y(2) (integer).
13. GATE 2025 ST Q45 (2 marks, Multiple choice) – Five iid bivariate N₂(0,I₂) vectors form rows of a 5×2 matrix X. Let W=XᵀX and Z=W₁₁+4W₁₂+4W₂₂. Find Var Z.
14. GATE 2025 ST Q53 (2 marks, Multiple select) – X~N₃(0,diag(4,9,4)),α=(2,0,−1)ᵀ,β=(1,1,1)ᵀ. Which hold?
15. GATE 2025 ST Q65 (2 marks, Numerical answer) – Bivariate normal X,Y have means 2,3,variances 16,25 and covariance 14. Find 2π[P(X>2,Y>3)−1/4] to two decimals.
16. GATE 2026 ST Q35 (1 mark, Numerical answer) – (X₁,X₂,X₃)ᵀ is normal with mean (1,1,1)ᵀ and covariance =pmatrix2&1&1\\1&2&1\\1&1&2pmatrix. Find Var(X₁ X₂=1,X₃=−1) to two decimals.
17. GATE 2026 ST Q47 (2 marks, Multiple choice) – Independent iid X,Y follow N p(μ,Σ),Σ positive definite. Which follows a chi-square law as stated?
18. GATE 2026 ST Q58 (2 marks, Multiple select) – X~N p(μ,Σ),Σ positive definite, and real A satisfies AᵀA=I. Let Y=AX. Which hold?
19. GATE 2026 ST Q59 (2 marks, Multiple select) – For X=(X₁,…,X p)ᵀ~N p(0,Σ),Σ positive definite, which hold?