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Computation of the Integral of the Bivariate Normal Distribution Over Arbitrary Polygons

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An efficient automatic procedure is given for evaluating the integral of the bivariate normal density function IBND over an arbitrary polygon Pi. The polygon Pi, defined by N points, falls into one or more of the following classes S, simple polygons S-bar, limit elements of sequences of uniformly bounded N-sided simple polygons of the same orientation Pi, arbitrary polygons, including self-intersecting SI ones, where S coincides with or belongs to S-bar which coincides with or belongs to Pi. It is not necessary to specify the class beforehand. The method extracts from Pi a set of N exterior angular regions. The IBND is evaluated over each of these, and the results are properly combined to yield IBND for Pi. In case Pi is SI, account must be taken of the number of its primary circuits and their orientations. A by-product of the analyses is the evaluation of a function APi. Another procedure for obtaining the same final results is described for completeness which is not as efficient. It treats an SI polygon by decomposing it into a finite set of S or S-bar type elements. The IBND is evaluated over each of these the results are properly summed to give the IBND for Pi. In contrast to the first method, the smallest class S, S-bar, Pi to which Pi belongs must be specified for computational efficiency. The Fortran IV programs for both procedures are presently set to yield approximately 3, 6, or 9-decimal-digit accuracy. Fortran IV listings of the programs are given.

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  • Statistics and Probability
  • Computer Programming and Software

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