Multivariate Gaußsche Verteilung
A multivariate Gaußsche Verteilung, also known as a multivariate Normalverteilung, is a generalization of the one-dimensional Normalverteilung to höhere Dimensionen. It describes the behavior of a vector of correlated random variables. This distribution is characterized by a mean vector and a Kovarianzmatrix.
The mean vector indicates the expected values of each variable in the distribution, while the covariance matrix captures the relationships between the variables, detailing how they vary together. Specifically, if we have a vector X consisting of n variables, the multivariate Gaußsche Verteilung ausgedrückt werden als:
P(X) = (1 / (2π)^(n/2) |Σ|^(1/2)) * exp(-1/2 * (X - μ)ᵀ Σ⁻¹ (X - μ))
where μ is the mean vector, Σ ist die Kovarianzmatrix, und |Σ| ist die Determinante der Kovarianzmatrix.
In practical applications, the multivariate Gaussian is widely used in various fields such as statistics, machine learning, and finance. It is particularly useful for modeling phenomena where several interrelated factors influence outcomes, such as in predictive modeling and Clustering-Algorithmen.
Eine wichtige Eigenschaft der multivariaten Gaußschen Verteilung ist, dass jede lineare Kombination of its variables will also follow a Gaussian distribution. This property makes it a powerful tool in both theoretical studies and practical applications.