Bivariate Normal Distribution

TOMS462 is a FORTRAN90 library which evaluates the upper right tail of the bivariate normal distribution; that is, the probability that normal variables X and Y with correlation R will satisfy H <= X and K <= Y.

While the text of many ACM TOMS algorithms is available online through ACM: or NETLIB:, most of the early algorithms are not available. This is one of them. I typed it in.


value = bivnor ( ah, ak, r )
computes VALUE, the probability that two variables, X and Y related by a bivariate normal distribution with correlation R, satisfy AH <= X and AK <= Y.


TOMS462 is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

OWENS, a FORTRAN90 library which evaluates Owen's T function.

PROB, a FORTRAN90 library which contains a number of routines for evaluating cumulative distribution functions.

TEST_VALUES, a FORTRAN90 library which supplies test values of various mathematical functions.


  1. Thomas Donnelly,
    Algorithm 462: Bivariate Normal Distribution,
    Communications of the ACM,
    October 1973, Volume 16, Number 10, page 638.
  2. Donald Owen,
    Tables for Computing Bivariate Normal Probabilities,
    Annals of Mathematical Statistics,
    Volume 27, Number 4, pages 1075-1090, December 1956.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN90 source codes.

Last revised on 13 April 2012.