Gaussian Distributions and the Heat Equation
In this chapter the Gaussian distribution is defined and its properties are explored. The chapter starts with the definition of a Gaussian distribution on the real line. In the process of exploring the properties of the Gaussian on the line, the Fourier transform and heat equation are introduced, and their relationship to the Gaussian is developed. The Gaussian distribution in multiple dimensions is defined, as are clipped and folded versions of this distribution. Some concepts from probability and statistics such as mean, variance, marginalization, and conditioning of probability densities are introduced in a concrete way using the Gaussian as the primary example. The properties of the Gaussian distribution are fundamental to understanding the concept of white noise, which is the driving process for all of the stochastic processes studied in this book. The main things to take away from this chapter are: To become familiar with the Gaussian distribution and its properties, and to be comfortable in performing integrals involving multi-dimensional Gaussians; To become acquainted with the concepts of mean, covariance, and informationtheoretic entropy; To understand how to marginalize and convolve probability densities, to compute conditional densities, and to fold and clip Gaussians; To observe that there is a relationship between Gaussian distributions and the heat equation; To know where to begin if presented with a diffusion equation, the symmetries of which are desired.
KeywordsHeat Equation Real Line Multivariate Gaussian Distribution Marginal Density Symmetry Analysis
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