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The bookUnit 4 · Linear Models and Multivariate81 / 83

Multivariate normal distribution

Why this is asked: Linear combinations of a multivariate normal are normal — that single fact plus the Wishart quadratic-form rule answers most questions here.

Multivariate normal and Wishart

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Zero correlation and independence: when they coincideinteractive

The equivalence holds under joint normality and fails without it — with the standard counterexample.

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The trap here

“If every marginal is normal then the vector is multivariate normal” — false

X ~ N(0,1) and Y=εXY = \varepsilon{}X with ε=±1\varepsilon = \pm1 independent

Both marginals are N(0,1) but X + Y is 0 half the time — not normal, so the pair is not jointly normal.

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Check yourself

Let X=(X1,X2)TX = (X_{1}, X_{2})^{T} be bivariate normal with mean (0, 0)ᵀ and covariance ∑=[[5,−3],[−3,10]]\sum = [[5, -3], [-3, 10]]. The mean vector and covariance matrix of Y=(X1,5−2X2)TY = (X_{1}, 5 - 2X_{2})^{T} are

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Open this in the full syllabus view · Unit 4