Multivariate normal: parameters and key properties.

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Answer

$X ~ N(μ, Σ), where μ is a length-d mean vector and Σ is a d\timesd covariance matrix (symmetric positive semi-definite).$
$( - 1 / 2)\; \cdot \;\operatorname{exp}( - 0.5\; \cdot \;(x - {\mu})\;{\Sigma}^{-1}\;(x - {\mu}))$
$Marginals and conditionals of a multivariate normal are also normal.$
$Linear combinations remain normal.$
$Uncorrelated components (Σ diagonal) \to independent.$
$Foundation of Gaussian processes, Kalman filters, LDA, and joint modeling.$

Check yourself — multiple choice

No structure
N(μ, Σ) with SPD Σ; marginals + conditionals + linear combos are normal; uncorrelated components independent
Only diagonal Σ
Same as Exp

MVN: closed under margins/conditionals/linear ops; uncorrelated ⇔ independent within MVN.

#distributions#probability

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