Probability density function of the multivariate t-distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate t-distribution ▷ Probability density function

Theorem: Let $X$ be a random vector following a multivariate t-distribution:

\[\label{eq:mvt} X \sim t(\mu, \Sigma, \nu) \; .\]

Then, the probability density function of $X$ is

\[\label{eq:mvt-pdf} f_X(x) = \sqrt{\frac{1}{(\nu \pi)^{n} |\Sigma|}} \, \frac{\Gamma([\nu+n]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{1}{\nu} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right]^{-(\nu+n)/2} \; .\]

Proof: This follows directly from the definition of the multivariate t-distribution.

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Sources:

original work

Metadata: ID: P333 | shortcut: mvt-pdf | author: JoramSoch | date: 2022-09-02, 11:50.