On the computation of the bivariate normal integral

@article{Drezner1990OnTC,
  title={On the computation of the bivariate normal integral},
  author={Zvi Drezner and George O. Wesolowsky},
  journal={Journal of Statistical Computation and Simulation},
  year={1990},
  volume={35},
  pages={101-107}
}
We propose a simple and efficient way to calculate bivariate normal probabilities. The algorithm is based on a formula for the partial derivative of the bivariate probability with respect to the correlation coefficient. 

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The bivariate normal distribution function may be expressed as the product of a marginal normal distribution times a conditional distribution. By approximating this conditional distribution, we

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A quadrature and two series representations are given as limiting cases of a bivariate t-distribution. The quadrature is taken over the complementary error function and the series are sums of Bessel

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This error will only have an impact if one of the ranges of integration is so extreme that the marginal probability of the corresponding coordinate being in the range of integration is less than

Mathematical functions and their approximations