GATE 2023 ST – Question 35

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 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.

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Correct answer: 0.35 to 0.40

Explanation

For a=(1,−2,2)ᵀ,the linear combination is centered normal with variance aᵀΣa=2+20+4−8+4−8=14. The event becomes |Z|<√[(7/2)/14]=1/2. Probability=1−2Φ(−.5)=1−.617=**0.38** after rounding.

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