GATE 2022 ST – Question 65

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 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.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 6 to 6

Explanation

The conditioning covariance is diag(3,2),and the cross covariance is (2,1). Normal conditional mean=3+(2,1)diag(1/3,1/2)(4−2,7−4)ᵀ=3+4/3+3/2=35/6≈5.833. The requested integer rounding gives **6**,matching the official accepted integer. The exact conditional expectation is 35/6,so the source’s “in integer” wording entails rounding here.

More Multivariate Normal and Wishart Distributions questions

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…
GATE 2026 ST Q58 – X~N p(μ,Σ),Σ positive definite, and real A satisfies AᵀA=I. Let Y=AX. Which hold?
GATE 2026 ST Q59 – For X=(X₁,…,X p)ᵀ~N p(0,Σ),Σ positive definite, which hold?

All Multivariate Normal and Wishart Distributions questions →