Multivariate Normal Distribution
X ~ N_p(μ, Σ)
Joint Elliptical-Contour Family • Published August 2, 2026
The joint generalization of the Normal distribution underlying ordinary linear regression and the multivariate trinity of tests: history from Galton and Pearson's correlation theory, an exact Schur-complement formula for conditional distributions verified via Bayes' rule to six decimal places, the Mahalanobis distance's exact tie to the already-published Chi-Square sheet, a rare case where zero covariance genuinely implies independence, and simulation via the Cholesky construction.
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About This Reference Sheet
The joint generalization of the Normal distribution underlying ordinary linear regression and the multivariate trinity of tests: history from Galton and Pearson's correlation theory, an exact Schur-complement formula for conditional distributions verified via Bayes' rule to six decimal places, the Mahalanobis distance's exact tie to the already-published Chi-Square sheet, a rare case where zero covariance genuinely implies independence, and simulation via the Cholesky construction.
Support
x ∈ ℝᵖ
Parameters
μ ∈ ℝᵖ (mean vector), Σ a p×p positive-definite covariance matrix