Definition: Multivariate normal distribution
Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Definition
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Metadata: ID: D1 | shortcut: mvn | author: JoramSoch | date: 2020-01-22, 05:20.
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.
- Koch KR (2007): "Multivariate Normal Distribution"; in: Introduction to Bayesian Statistics, ch. 2.5.1, pp. 51-53, eq. 2.195; URL: https://www.springer.com/gp/book/9783540727231; DOI: 10.1007/978-3-540-72726-2.
Metadata: ID: D1 | shortcut: mvn | author: JoramSoch | date: 2020-01-22, 05:20.