The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

The multivariate normal distribution, or multivariate Gaussian distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One possible definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.

If ${\bf X}\sim \text{MVN}({\bf \mu, \Sigma})$ and ${\bf x} \in \mathbb{R}^n$ then:

$$f(x) = \frac{1}{{(2\pi)}^{n/2}\sqrt{|{\bf \Sigma}|}}\exp[-\frac12 {\bf (x - \mu)^T\Sigma^{-1}(x - \mu)}]$$

history | excerpt history