The multivariate normal distribution

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stochastikclips

Published on Nov 29, 2022
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For every d greater than or equal to 2, the (non-degenerate) d-dimensional normal distribution arises as the distribution of an affine transformation AY + b of a random vector Y with stochastically independent and standard normally distributed components. Here, A is a d-row invertible matrix and b is a vector in d-dimensional Euclidean space. In this video, the non-degenerate d-dimensional normal distribution of a random vector X is defined in this (constructive) way. The density of X is determined using the transformation theorem for Lebesgue densities, see • The transformation theorem for Lebesgue... To watch the video you should know the one-dimensional normal distribution and you should be familiar with the terms expected value vector and covariance matrix, see • Expected value vector and covariance matrix It may also be helpful to first familiarize yourself with the special case d=2, see • The two-dimensional normal distribution The video shows the existence of d-dimensional normal distributions and makes it clear that the set of all non-degenerate d-dimensional normal distributions is closed with respect to affine transformations. Furthermore, it is proven that the (one-dimensional) marginal distributions of a normally distributed random vector are normal distributions and that in the case of a d-dimensional normal distribution, independence and uncorrelation are equivalent. Another point is the principal component representation of a multivariate normally distributed random vector. Finally, a definition of the multivariate normal distribution is given, which includes the case of degenerate normal distributions.

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