GATE 2025 ST – Question 53

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

X~N₃(0,diag(4,9,4)),α=(2,0,−1)ᵀ,β=(1,1,1)ᵀ. Which hold?

E trace(XXᵀααᵀ)=20
Var trace(Xαᵀ)=20
E trace(XXᵀ)=17
Cov(αᵀX,βᵀX)=3

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Correct answer: (A) E trace(XXᵀααᵀ)=20; (B) Var trace(Xαᵀ)=20; (C) E trace(XXᵀ)=17

Explanation

The first trace is (αᵀX)² and has expectation αᵀΣα=4·4+4=20. The second trace is αᵀX, with variance the same 20. E trace(XXᵀ)=traceΣ=4+9+4=17. The covariance is αᵀΣβ=8−4=4, not 3. **A,B,C**.

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