Computing the bivariate Gaussian probability integral

摘要

In signal processing applications, it is often required to computenthe integral of the bivariate Gaussian probability density functionn(PDF) over the four quadrants. When the mean of the random variables arennonzero, computing the closed form solution to these integrals with thenusual techniques of integration is infeasible. Many numerical solutionsnhave been proposed; however, the accuracy of these solutions depends onnvarious constraints. In this work, we derive the closed form solution tonthis problem using the characteristic function method. The solution isnderived in terms of the well-known confluent hypergeometric function.nWhen the mean of the random variables is zero, the solution is shown tonreduce to a known result for the value of the integral over the firstnquadrant. The solution is implementable in software packages such asnMAPLE