Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Definition

Definition: Let $X$ be an $n \times 1$ random vector. Then, $X$ is said to be multivariate normally distributed with mean $\mu$ and covariance $\Sigma$

\[\label{eq:mvn} X \sim \mathcal{N}(\mu, \Sigma) \; ,\]

if and only if its probability density function is given by

\[\label{eq:mvn-pdf} \mathcal{N}(x; \mu, \Sigma) = \frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right]\]

where $\mu$ is an $n \times 1$ real vector and $\Sigma$ is an $n \times n$ positive definite matrix.


Metadata: ID: D1 | shortcut: mvn | author: JoramSoch | date: 2020-01-22, 05:20.