Vektfordeling

Vektfordeling, or vector distribution, describes the joint distribution of a random vector X = (X1, ..., Xk). It assigns probabilities or densities to vectors in R^k. For discrete components, the joint distribution is given by a probability mass function p(x1, ..., xk); for continuous components, a joint probability density function f(x) exists with P(a ≤ X ≤ b) = ∫_a^b f(x) dx. The joint cumulative distribution function F(x) = P(X1 ≤ x1, ..., Xk ≤ xk) also summarizes the distribution.

The distribution is commonly described by the mean vector μ = E[X] and the covariance matrix Σ = Cov(X). Marginal distributions of individual components follow from the margins of the joint distribution. Components may be independent (the joint distribution factors into the product of marginals) or merely uncorrelated; correlation measures linear dependence between components.

A central example is the multivariate normal distribution, defined by μ and Σ, with density f(x) ∝ exp(-1/2 (x-μ)^T Σ^{-1} (x-μ)). Other examples include the multivariate t distribution, Dirichlet distributions on simplices, and copula-based models that combine specified margins with flexible dependence structures.

Transformations are a key feature: for a linear transformation Y = AX + b, the distribution of Y follows from that of X; if X is multivariate normal, then Y is also multivariate normal with mean Aμ + b and covariance AΣA^T. Marginalization and conditioning yield distributions for sub-vectors; conditioning is particularly tractable in the normal case. Copulas separate margins from dependence in modeling vektfordelinger.

Applications span statistics, finance, engineering and data science, where modeling the joint behavior of several variables supports inference, prediction and risk assessment.