Multivariate t Distribution
X ~ t_p(ν, μ, Σ)
Joint Heavy-Tailed / Elliptical-Contour Family • Published July 22, 2026
The direct joint generalization of the already-published Student's t, built by dividing an already-published Multivariate Normal by a single shared Chi-Square scale (confirmed numerically to match scipy.stats.multivariate_t exactly, and to converge to the Multivariate Normal as ν grows). Every marginal is a univariate t sharing the same degrees of freedom -- verified via a Kolmogorov-Smirnov test -- and the covariance is always larger than the scale matrix Σ itself, exactly as in the univariate case.
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About This Reference Sheet
The direct joint generalization of the already-published Student's t, built by dividing an already-published Multivariate Normal by a single shared Chi-Square scale (confirmed numerically to match scipy.stats.multivariate_t exactly, and to converge to the Multivariate Normal as ν grows). Every marginal is a univariate t sharing the same degrees of freedom -- verified via a Kolmogorov-Smirnov test -- and the covariance is always larger than the scale matrix Σ itself, exactly as in the univariate case.
Support
x ∈ ℝᵖ
Parameters
ν > 0 (degrees of freedom), μ ∈ ℝᵖ (location), Σ a p×p positive-definite scale matrix