Merge pull request #173 from JoramSoch/master

added 1 proof

Author: JoramSoch

I/ToC.md

@@ -453,7 +453,8 @@ title: "Table of Contents"
 
    4.2. Multivariate t-distribution <br>
    &emsp;&ensp; 4.2.1. *[Definition](/D/mvt)* <br>
-   &emsp;&ensp; 4.2.2. **[Relationship to F-distribution](/P/mvt-f)** <br>
+   &emsp;&ensp; 4.2.2. **[Probability density function](/P/mvt-pdf)** <br>
+   &emsp;&ensp; 4.2.3. **[Relationship to F-distribution](/P/mvt-f)** <br>
 
    4.3. Normal-gamma distribution <br>
    &emsp;&ensp; 4.3.1. *[Definition](/D/ng)* <br>

P/mvt-pdf.md

@@ -0,0 +1,37 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-09-02 11:50:00
+
+title: "Probability density function of the multivariate t-distribution"
+chapter: "Probability Distributions"
+section: "Multivariate continuous distributions"
+topic: "Multivariate t-distribution"
+theorem: "Probability density function"
+
+sources:
+
+proof_id: "P333"
+shortcut: "mvt-pdf"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a [random vector](/D/rvec) following a [multivariate t-distribution](/D/mvt):
+
+$$ \label{eq:mvt}
+X \sim t(\mu, \Sigma, \nu) \; .
+$$
+
+Then, the [probability density function](/D/pdf) 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](/D/mvt).