GATE 2020 ST – Question 27

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 1 mark · Numerical answer

X∼N₄(0,Σ), with $$\Sigma=\begin{pmatrix}1&.7&.6&.4\\.7&1&.5&.8\\.6&.5&1&.7\\.4&.8&.7&1\end{pmatrix}.$$ Find P(X₁+X₂+X₃+X₄>0) (one decimal).

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Correct answer: 0.5 to 0.5

Explanation

Any linear combination of this centred multivariate normal vector is centred normal. The sum has positive variance, so its continuous distribution is symmetric about zero. Consequently P(sum>0)=1/2=0.5; the individual covariance values affect variance but not this symmetry probability.

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