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


Authors: Hatem A. Fayed and Amir F. Atiya
Journal: Math. Comp. 83 (2014), 235-250
MSC (2010): Primary 33B20, 33C45
DOI: https://doi.org/10.1090/S0025-5718-2013-02720-2
Published electronically: May 29, 2013
MathSciNet review: 3120588
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Abstract: This paper derives the value of the integral of the product of the error function and the normal probability density as a series of the Hermite polynomial and the normalized incomplete Gamma function. This expression is beneficial, and can be used for evaluating the bivariate normal integral as a series expansion. This expansion is a good alternative to the well-known tetrachoric series, when the correlation coefficient, $ \rho $, is large in absolute value.


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Additional Information

Hatem A. Fayed
Affiliation: Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
Email: h{\textunderscore}fayed@eng.cu.edu.eg

Amir F. Atiya
Affiliation: Department of Computer Engineering, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
Email: amir@alumni.caltech.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02720-2
Keywords: Error function, normal probability, Gamma function, Hermite polynomial, hypergeometric function, bivariate normal integral, tetrachoric series
Received by editor(s): October 5, 2011
Received by editor(s) in revised form: February 20, 2012
Published electronically: May 29, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.