Graphs for Bivariate Normal Probabilities

Abstract

1. Introduction and summary. Recently there has been much activity dealing with the tabulation of the bivariate normal probability integral. D. B. Owen [3], [4] has summarized many of the properties of the bivariate normal distribution function and tabulated an auxiliary function which enables one to calculate the bivariate normal probability integral. In addition, the National Bureau of Standards [1] has compiled extensive tables of the bivariate normal integral drawn from the works of K. Pearson, Evelyn Fix and J. Neymaln, and H. H. Germond. In this same volume, D. B. Owen has contributed an extensive section on applications. It is the purpose of this paper to present three charts, which will enable one to easily compute the bivariate normal integral to a maximum error of 10-2. This should be sufficient for most practical applications. Owen and Wiesen [5] have also presented charts with a similar objective; however, as pointed out below, we believe the charts presented here lend themselves more easily to visual interpolation. Actually the motivation for these charts came from the Owen and Wiesen work. 2. Notation and formulas. We present here notation and useful formulas relating to the bivariate normal integral. Let X and Y be random variables following a bivariate normal distribution with zero means, unit variances, and correlation coefficient p. Then