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GATE 2026 ST – Question 59

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

For X=(X₁,…,X_p)ᵀ~N_p(0,Σ),Σ positive definite, which hold?

  1. All coordinates are always independent normals.
  2. Each coordinate is normal.
  3. Σ_iX_i² always has a chi-square distribution.
  4. Every linear combination of the coordinates is normal.

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Show answer and explanation

Correct answer: (B) Each coordinate is normal.; (D) Every linear combination of the coordinates is normal.

Explanation

Normality of all linear combinations characterizes a multivariate normal vector, and individual coordinates are special cases, so B,D hold (including a degenerate zero combination). Correlated coordinates need not be independent. The sum of raw squares is χ²_p only with the required standardized covariance, whereas XᵀΣ⁻¹X always is χ²_p. **B,D**.

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