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chi-squared distribution

Cut-off BVN limited to an ellipsoidal core

Covariance matrix of a cut-off Multivariate Normal Distribution – III – results for a 2-dimensional BVN-core and proper normalization of its cut-off distribution

In the math section of this blog, we try to cover interesting aspects of Multivariate Normal Distributions [MVNs]. The topic of this post series is the covariance matrix of a MVN-like distribution confined inside a hyper-surface of constant probability density. Outside of the surface we set the probability density to zero. This gives us a “cut-off” MVN- distribution. Contour surfaces… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – III – results for a 2-dimensional BVN-core and proper normalization of its cut-off distribution

3-dim projections of 4-dim MVN

Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

In the 1st post of this series, we have posed the following problem: Take the probability density of a Multivariate Normal Distribution [MVN], but set it to zero at Mahalanobis distances bigger than a finite distance D. Take a respective volume region enclosed by a contour surface of constant probability density (at the Mahalanobis distance dm=D). We have called such… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

Confidence ellipses for an approximate BVD

Bivariate Normal Distribution – integrated probability up to a given Mahalanobis distance, the Chi-squared distribution and confidence ellipses

In previous posts of this blog we have discussed the general form of the probability density function [pdf] of a Bivariate Normal Distribution [BVD]. In this post we consider the integral over a BVD’s pdf up to a defined value of the Mahalanobis Distance. A given value of the latter defines an elliptic contour line of constant probability density. With… Read More »Bivariate Normal Distribution – integrated probability up to a given Mahalanobis distance, the Chi-squared distribution and confidence ellipses