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probability density function

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

Bivariate Normal Distribution

Bivariate normal distribution – derivation by linear transformation of a random vector of two independent Gaussians

In an another post on properties of a Bivariate Normal Distribution [BVD] I have motivated the form of its probability density function [pdf] by symmetry arguments and the underlying probability density functions of its marginals, namely 1-dimensional Gaussians. In this post we will derive the probability density function by following the line of argumentation for a general Multivariate Normal Distribution… Read More »Bivariate normal distribution – derivation by linear transformation of a random vector of two independent Gaussians

Concentric surfaces of ellipsoids

Multivariate Normal Distributions – III – Variance-Covariance Matrix and a distance measure for vectors of non-degenerate distributions

In previous posts of this series I have motivated the functional form of the probability density of a so called “non-degenerate Multivariate Normal Distribution“. In this post we will have a closer look at the matrix Σ that controls the probability density function [pdf] of such a distribution. We will show that it actually is the covariance matrix of the… Read More »Multivariate Normal Distributions – III – Variance-Covariance Matrix and a distance measure for vectors of non-degenerate distributions

Linear transformed 3-dim Z-distribution

Multivariate Normal Distributions – II – Linear transformation of a random vector with independent standardized normal components

In Machine Learning we typically deal with huge, but finite vector distributions defined in the ℝn. At least in certain regions of the ℝn these distributions may approximate an underlying continuous distribution. In the first post of this series we worked with a special type of continuous vector distribution based on independent 1-dimensional standardized normal distributions for the vector components.… Read More »Multivariate Normal Distributions – II – Linear transformation of a random vector with independent standardized normal components