{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:39:35Z","timestamp":1776843575980,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"285","license":[{"start":{"date-parts":[[2014,5,29]],"date-time":"2014-05-29T00:00:00Z","timestamp":1401321600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    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,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is large in absolute value.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02720-2","type":"journal-article","created":{"date-parts":[[2013,5,29]],"date-time":"2013-05-29T15:13:40Z","timestamp":1369840420000},"page":"235-250","source":"Crossref","is-referenced-by-count":17,"title":["An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral"],"prefix":"10.1090","volume":"83","author":[{"given":"Hatem","family":"Fayed","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amir","family":"Atiya","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,5,29]]},"reference":[{"key":"1","unstructured":"[Abramowitz and Stegun(1964)] M. 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Briggs, http:\/\/keithbriggs.info\/documents\/erf-integrals.pdf, 2003."},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/5436","volume-title":"Quantitative finance and risk management","author":"Dash, Jan W.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/9812387129"},{"issue":"4","key":"6","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1214\/aos\/1176344739","article-title":"Calculation of univariate and bivariate normal probability functions","volume":"7","author":"Divgi, D. R.","year":"1979","journal-title":"Ann. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0090-5364","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[Donnelly(1973)] T.G. Donnelly, Algorithm 462: Bivariate normal distribution, Commun. 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Econ. 110 (4) (1995), 941\u2013974.","DOI":"10.2307\/2946645"},{"key":"11","unstructured":"[Gai(2002)] J. Gai, A computational study of the bivariate normal probability function, M.Sc. thesis, Department of Mathematics and Statistics, Queen\u2019s University, Kingston, Ontario, Canada, 2002."},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1023\/B:STCO.0000035304.20635.31","article-title":"Numerical computation of rectangular bivariate and trivariate normal and \ud835\udc61 probabilities","volume":"14","author":"Genz, Alan","year":"2004","journal-title":"Stat. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0960-3174","issn-type":"print"},{"key":"13","series-title":"Wiley Series in Probability and Mathematical Statistics","volume-title":"Distributions in statistics: continuous multivariate distributions","author":"Johnson, Norman L.","year":"1972"},{"key":"14","doi-asserted-by":"crossref","unstructured":"[Martzoukos(2001)] S.H. Martzoukos, The option on \ud835\udc5b assets with exchange rate and exercise price risk. J. Multinatl. Financ. Manage. 11 (1) (2001), 1\u201315.","DOI":"10.1016\/S1042-444X(00)00039-6"},{"key":"15","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1093\/biomet\/33.1.59","article-title":"The probability integral for two variables","volume":"33","author":"Nicholson, C.","year":"1943","journal-title":"Biometrika","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3444","issn-type":"print"},{"key":"16","doi-asserted-by":"publisher","first-page":"1075","DOI":"10.1214\/aoms\/1177728074","article-title":"Tables for computing bivariate normal probabilities","volume":"27","author":"Owen, Donald B.","year":"1956","journal-title":"Ann. Math. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4851","issn-type":"print"},{"key":"17","doi-asserted-by":"crossref","unstructured":"[Pearson(1901)] K. Pearson, Mathematical contributions to the theory of evolution. VII. on the correlation of characters not quantitatively. Philos. Trans. R. Soc. S-A. 196 (1901), 1\u201347.","DOI":"10.1098\/rsta.1900.0022"},{"key":"18","doi-asserted-by":"crossref","unstructured":"[Pearson(1903)] K. Pearson, Mathematical contributions to the theory of evolution. XI. on the influence of natural selection on the variability and correlation of organs. Philos. Trans. R. Soc. S-A. 200 (1903), 1\u201366.","DOI":"10.1098\/rsta.1903.0001"},{"key":"19","unstructured":"[Sheppard(1900)] W.F. Sheppard, On the calculation of the double-integral expressing normal correlation, Trans. Camb. Philos. Soc. 19 (1900), 23\u201366."},{"key":"20","doi-asserted-by":"crossref","unstructured":"[Simon and Divsalar(1998)] M.K. Simon, D. Divsalar, Some new twists to problems involving the Gaussian probability integral, IEEE Trans. Commun. 46 (2) (1998), 200\u2013210.","DOI":"10.1109\/26.659479"},{"key":"21","doi-asserted-by":"crossref","unstructured":"[Terza and Welland(1991)] J. Terza, U. 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