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variance-covariance matrix

Contour ellipses from Cholesky decomp ot various covariance matrices

Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle

In my last post about Bivariate Normal Distributions [BVD] I have discussed why contour lines of a BVD’s probability density function [pdf] are concentric ellipses. These contour ellipses are defined by constant values of the so called Mahalanobis distance. In addition, I have discussed a method to create these ellipses from values of the elements of the BVD’s (variance-) covariance… Read More »Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle

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

contour ellipsoids of a projected MND

Multivariate Normal Distributions – I – Basics and a random vector of independent Gaussians

This post series is about mathematical aspects of so called “Multivariate Normal Distributions“. In respective literature two abbreviations are common: MNDs or MVNs. I will use both synonymously. To get an easy access, I want to introduce a MND as the result of a linear transformations applied to random vectors whose components can be described by independent 1-dimensional normal distributions.… Read More »Multivariate Normal Distributions – I – Basics and a random vector of independent Gaussians

Probability density function of a Bivariate Normal Distribution – derived from assumptions on marginal distributions and functional factorization

For a better understanding of ML experiments regarding a generator of human faces based on a convolutional autoencoder we need an understanding of multivariate and bivariate normal distributions and their probability densities. This post is about the probability density function [pdf] of a bivariate normal distribution of two correlated Gaussian random variables X and Y. Most derivations of the mathematical… Read More »Probability density function of a Bivariate Normal Distribution – derived from assumptions on marginal distributions and functional factorization