  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.
