Multivariate normal distribution

A probability distribution for a vector of jointly normal random variables, determined by its mean vector and covariance matrix. Its linear projections are normally distributed, and its covariance matrix encodes dependence.

Normal distribution: A continuous probability distribution with a symmetric, bell-shaped density determined by its mean and variance. Each one-dimensional projection of a multivariate normal vector has this distribution.
Probability density function: A function whose values describe the relative density of a continuous probability distribution. For a nonsingular multivariate normal, it is an exponential of a covariance-weighted squared distance.
Multivariate linear regression: A regression model that predicts multiple response variables jointly from explanatory variables. Its error vectors are often modeled as multivariate normal to capture correlated outcomes.
Independent random variables: Random variables whose joint probabilities factor into products of their marginal probabilities. Normal components need not be independent; for jointly normal variables, zero covariance does imply independence.
Random vector: A vector whose components are random variables. The distribution describes the joint behavior of all components at once.
Mahalanobis distance: A distance from a point to a distribution center that accounts for covariance and scale. Its squared value forms the exponent in the multivariate normal density.
Kalman filter: An algorithm that estimates a changing hidden state from noisy observations in a linear system. It propagates and updates Gaussian state distributions using linear transformations and conditioning.
Elliptical distribution: A family of multivariate distributions with contours shaped as ellipsoids under suitable conditions. The multivariate normal is a prominent member, distinguished by normal linear projections.
Covariance matrix: A matrix whose entries are covariances between pairs of random variables. It specifies component variances and pairwise dependence in the distribution.
Schur complement: A matrix derived from a block matrix by eliminating one diagonal block. It gives the conditional covariance when a jointly normal vector is partitioned.

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