A reduction formula for normal multivariate integrals
Biometrika (1954)

MR Let (X1,X2,⋯,Xn) be a vector of chance variables with a nonsingular multivariate normal distribution. The problem is to evaluate P(X1>a1,⋯,Xn>an). The author obtains a reduction formula for this probability, involving integrals of partial derivatives of the probability with respect to the elements of the covariance matrix of (X1,⋯,Xn). For n=3 and n=4, the reduction formula enables the author to express the probability as a finite sum of single integrals of tabulated functions. These integrals have to be evaluated by numerical quadrature, but for certain cases simple approximations to them are given.
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