MathLiveMultivariate normal distribution density
Gives the probability density of an n-dimensional normal (Gaussian) random vector with mean vector mu and covariance matrix Sigma. It is used throughout multivariate statistics, for example in modelling correlated measurements and in Bayesian inference.
f(\boldsymbol{x}) = (2 \pi)^{-n/2} (\det \boldsymbol{\Sigma})^{-1/2} \mathrm{e}^{-\frac{1}{2}(\boldsymbol{x} - \boldsymbol{\mu})^{\mathrm{T}} \boldsymbol{\Sigma}^{-1} (\boldsymbol{x} - \boldsymbol{\mu})}What the symbols mean
| probability density at the point \boldsymbol{x} | |
| random vector of n variables at which the density is evaluated | |
| dimension of the vector \boldsymbol{x} | |
| mean (location) vector | |
| positive definite covariance matrix | |
| determinant of the covariance matrix | |
| inverse of the covariance matrix | |
| transpose of the vector \boldsymbol{x} |
About this formula
- FieldStatistics
- TopicProbability distributions
- LevelUniversity
- Also calledmultivariate Gaussian density, joint normal distribution pdf, multivariate normal pdf, multivariate Gaussian distribution formula, n-dimensional normal density, density of multivariate normal
- SourceWikidata Q1149000
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