The GATE Grind

GATE 2026 ST – Question 58

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

X~N_p(μ,Σ),Σ positive definite, and real A satisfies AᵀA=I. Let Y=AX. Which hold?

  1. Y~N_p(Aμ,AᵀΣA)
  2. Y~N_p(Aμ,AΣAᵀ)
  3. Var(Σ_iY_i)=Var(Σ_iX_i)
  4. Σ_iVar(Y_i)=Σ_iVar(X_i)

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) Y~N_p(Aμ,AΣAᵀ); (D) Σ_iVar(Y_i)=Σ_iVar(X_i)

Explanation

A linear normal transformation gives mean Aμ and covariance AΣAᵀ, proving B. Orthogonality preserves trace: trace(AΣAᵀ)=trace(ΣAᵀA)=traceΣ, proving D. It does not generally preserve the particular all-ones direction’s variance in C. For example with Σ=diag(1,2), rotating that direction onto a coordinate axis changes its variance. A has the covariance factors in the wrong order. **B,D**.

More Multivariate Normal and Wishart Distributions questions

All Multivariate Normal and Wishart Distributions questions →