Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Normal-gamma distribution ▷ Definition

Definition: Let $X$ be an $n \times 1$ random vector and let $Y$ be a positive random variable. Then, $X$ and $Y$ are said to follow a normal-gamma distribution

\[\label{eq:ng} X,Y \sim \mathrm{NG}(\mu, \Lambda, a, b) \; ,\]

if and only if their joint probability density function is given by

\[\label{eq:ng-pdf} f_{X,Y}(x,y) = \mathcal{N}(x; \mu, (y \Lambda)^{-1}) \cdot \mathrm{Gam}(y; a, b)\]

where $\mathcal{N}(x; \mu, \Sigma)$ is the probability density function of the multivariate normal distribution with mean $\mu$ and covariance $\Sigma$ and $\mathrm{Gam}(x; a, b)$ is the probability density function of the gamma distribution with shape $a$ and rate $b$. The $n \times n$ matrix $\Lambda$ is referred to as the precision matrix of the normal-gamma distribution.


Metadata: ID: D5 | shortcut: ng | author: JoramSoch | date: 2020-01-27, 14:28.