# 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.

## 216 Citations

### A simple approximation for the bivariate normal integral

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### Computation of the multivariate normal integral

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### Numerical Computation of Multivariate Normal Probabilities

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This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms.

### Trapped by the Tails of the Bivariate Normal Distribution

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Through the example of partial barrier options, we show that accuracy in the tail of the bivariate normal distribution is critical. We then propose a small change to a popular algorithm for the…

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The new approximation method is compared with approximation methods based on products of univariate normal probabilities, using tests with random covariance-matrix/probability-region problems for up to twenty variables.

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This work provides various (equivalent) expressions for the bivariate normal copula, its Gini's gamma is computed, and improved bounds and approximations on its diagonal are derived.

### Numerical computation of rectangular bivariate and trivariate normal and t probabilities

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Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations, and a generalization of Plackett's formula is derived for bivariate and trivariate t probabilities.

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

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- 2014

This expression is beneficial, and can be used for evaluating the bivariate normal integral as a series expansion, a good alternative to the well-known tetrachoric series, when the correlation coefficient, ρ, is large in absolute value.

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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…