An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral (Q2862527)

The authors study the integral NEWLINE\[NEWLINEI(a,b,x):=\int_0^xe^{-t^2}\int_0^{at+b}e^{-s^2}\,ds\,dt=\frac{\sqrt{\pi}}{2}\int_0^xe^{-t^2}\text{erf}(at+b)\,dt.NEWLINE\]NEWLINE This integral can be written as an infinite series of Hermite polynomials and the normalized incomplete gamma function. This expression is used to derive series expansions for the bivariate normal integral. The fast convergence of these expansions leads to a comparison with other numerical methods for the bivariate normal complementary integral -- the Pearson and the Vasicek method. It turns out that the new expansion is a good alternative to the well-known tetrachoric series, when the correlation coefficient is large in absolute value.