Multivariate Normal Distribution

Recall that the univariate normal distribution is described by the function

$$p(x|\mu ,\sigma )=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$$

The multivariate normal distribution is a generalization of the univariate normal distribution for a multidimensional random variable:

$$p(\mathbf{x}|\mu ,\mathbf{\Sigma })=\frac{1}{\sqrt{(2\pi {)}^n|\mathbf{\Sigma }|}}\exp \left(-\frac{1}{2}(\mathbf{x}-\mu {)}^T{\mathbf{\Sigma }}^{-1}(\mathbf{x}-\mu )\right)$$

where:

• $\mathbf{x}$ is an $n$-dimensional random vector
• $\mu$ is an $n$-dimensional mean vector
• $\mathbf{\Sigma }$ is a square $n\times n$ covariance matrix

Bivariate Distribution

The projection of the 3-dimensional Gaussian slice is an ellipse.