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A Diffusion Equation for the Density of the Ratio of Two Jointly Distributed Gaussian Variables and the Exponential Analysis Problem

It is shown that the density of the ratio of two random variables with the same variance and joint Gaussian density satisfies a nonstationary diffusion equation. Implications of this result for adaptive kernel density estimation of the condensed density of the generalized eigenvalues of a random matrix pencil useful for solving the exponential analysis problem are discussed.