MultivariateNormalverteilung

The Multivariate Normal Distribution, often abbreviated as MVN, is a generalization of the one-dimensional normal distribution to higher dimensions. It is a probability distribution for a vector of random variables, each of which has a normal distribution. The distribution is specified by a mean vector and a covariance matrix.

The probability density function of an MVN with mean vector μ and covariance matrix Σ is given by:

f(x) = (2π)^(-k/2) |Σ|^(-1/2) exp(-(1/2)(x - μ)^T Σ^(-1)(x - μ))

where x is a k-dimensional vector, μ is the k-dimensional mean vector, Σ is the k×k covariance matrix, |Σ| is the determinant of Σ, and T denotes the transpose of a vector.

The MVN is widely used in statistics and machine learning due to its mathematical tractability and the fact that it is the maximum entropy probability distribution for a given mean and covariance. It is also used as a prior distribution in Bayesian statistics.

The MVN has several important properties, including:

The marginal distribution of any subset of the variables is also multivariate normal.

The conditional distribution of a subset of the variables, given the values of the other variables, is also multivariate normal.

The sum of two independent multivariate normal random variables is also multivariate normal.

The MVN is often used in hypothesis testing, regression analysis, and other statistical inference procedures. It is also used in machine learning for tasks such as Gaussian process regression and as a component of more complex probabilistic models.