A multivariate normal distribution generalizes the ordinary normal distribution to a random vector.

It is characterized by a mean vector $\boldsymbol\mu$ and covariance matrix $\Sigma$:

$$\mathbf X\sim\mathcal N(\boldsymbol\mu,\Sigma).$$

Elliptical uncertainty

For a nondegenerate distribution, equal-density contours satisfy approximately

$$(\mathbf x-\boldsymbol\mu)^T\Sigma^{-1}(\mathbf x-\boldsymbol\mu)=\text{constant}.$$

In two dimensions these contours are ellipses. Their principal directions are given by covariance eigenvectors, while eigenvalues control the spread along those directions.

Marginals and linear combinations

Every linear combination

$$\mathbf a^T\mathbf X$$

is normally distributed. Marginal distributions obtained by selecting components are also multivariate normal.

Independence and covariance

For jointly Gaussian variables, zero covariance implies independence. This is stronger than the general case, where uncorrelated variables need not be independent.

Why Gaussian models are convenient

Linear transformations preserve Gaussian form, and conditioning a joint Gaussian distribution produces another Gaussian distribution. These closure properties make Gaussian uncertainty especially convenient for linear state estimation.