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

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

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

Then, the marginal distribution of any $m$-dimensional subset vector $X_s$ is also a multivariate t-distribution

\[\label{eq:mvt-marg} X_s \sim t(\mu_s, \Sigma_s, \nu)\]

where $\mu_s$ drops the irrelevant variables (the ones not in the subset, i.e. marginalized out) from the mean vector $\mu$ and $\Sigma_s$ drops the corresponding rows and columns from the covariance matrix $\Sigma$.

Proof: Define an $m \times n$ subset matrix $S$ such that $s_{ij} = 1$, if the $j$-th element in $X_s$ corresponds to the $i$-th element in $X$, and $s_{ij} = 0$ otherwise. Then,

\[\label{eq:xs} X_s = S X\]

and we can apply the linear transformation theorem to give

\[\label{eq:mvt-marg-qed} X_s \sim t(S \mu, S \Sigma S^\mathrm{T}, \nu) \; .\]

Finally, we see that $S \mu = \mu_s$ and $S \Sigma S^\mathrm{T} = \Sigma_s$.

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Metadata: ID: P525 | shortcut: mvt-marg | author: JoramSoch | date: 2026-01-23, 14:04.