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Pearson correlation coefficient

Contours of a bivariate normal distribution

Bivariate normal distribution – explicit reconstruction of a BVD random vector via Cholesky decomposition of the covariance matrix

In other posts of this blog I have discussed the general form of a Bivariate Normal Distribution [BVD] . For a centered Cartesian coordinate system [CCS] (see below), we have already seen the following: In this post I will give you a recipe to explicitly construct two random variables X, Y of a BVD from 1-dimensional Gaussians Z1, Z2 with… Read More »Bivariate normal distribution – explicit reconstruction of a BVD random vector via Cholesky decomposition of the covariance matrix

Bivariate Normal Distribution

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

Bivariate Normal Distribution from face data encoded by a CAE

Bivariate Normal Distribution – derivation of the covariance and correlation by integration of the probability density

In a previous post of this blog we have derived a function g2(x,y for the probability density f a Bivariate Normal Distribution [BVD] of two 1-dimensional random variables X and Y). By rewriting the probability density function [pdf] in terms of vectors (x, y)T and a coupling matrix Σ-1 we recognized that a coefficient appearing in a central exponential of… Read More »Bivariate Normal Distribution – derivation of the covariance and correlation by integration of the probability density

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