Exact Distribution for the Product of Two Correlated Gaussian Random Variables

This letter considers the distribution of product for two correlated real Gaussian random variables with nonzero means and arbitrary variances, which arises widely in radar and communication societies. We determine the exact probability density function (PDF) in terms of an infinite sum of modified Bessel functions of second kind, which includes some existent results, i.e., zero-means and/or independent variables, as special cases. Then, we study the approximation error and convergence rate when finite summations are exploited in practice. Finally, we evaluate the PDF behaviors of the derived expression as well as the Monte Carlo simulations.