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A Toolbox for the Radial and Angular Marginalization of Bivariate Normal Distributions

Bivariate normal distributions are often used to describe the joint probability density of a pair of random variables. These distributions arise across many domains, from telecommunications, to meteorology, ballistics, and computational neuroscience. In these applications, it is often useful to radially and angularly marginalize (i.e.,~under a polar transformation) the joint probability distribution relative to the coordinate system's origin. This marginalization is trivial for a zero-mean, isotropic distribution, but is non-trivial for the most general case of a non-zero-mean, anisotropic distribution with a non-diagonal covariance matrix. Across domains, a range of solutions with varying degrees of generality have been derived. Here, we provide a concise summary of analytic solutions for the polar marginalization of bivariate normal distributions. This report accompanies a Matlab (Mathworks, Inc.) and R toolbox that provides closed-form and numeric implementations for the marginalizations described herein.