For a more detailed description of the bivariate Normal distribution and its properties see Abramowitz and Stegun (1972) and Kendall and Stuart (1969). The method used is described by Genz (2004).

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and

t

probabilities Statistics and Computing 14 151–160

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

5 Arguments

1: $x$ – doubleInput: On entry: $x$ , the first argument for which the bivariate Normal distribution function is to be evaluated.
2: $y$ – doubleInput: On entry: $y$ , the second argument for which the bivariate Normal distribution function is to be evaluated.
3: $rho$ – doubleInput: On entry: $ρ$ , the correlation coefficient.

Constraint: $- 1.0 \leq rho \leq 1.0$ .
4: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

On any of the error conditions listed below nag_bivariate_normal_dist (g01hac) returns $0.0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT: On entry, $rho = 〈value〉$ .
Constraint: $rho \leq 1.0$ .
NE_REAL_ARG_LT: On entry, $rho = 〈value〉$ .
Constraint: $rho \geq - 1.0$ .

Accuracy of the hybrid algorithm implemented here is discussed in Genz (2004). This algorithm should give a maximum absolute error of less than

5 \times 10^{- 16}

nag_bivariate_normal_dist (g01hac) is not threaded in any implementation.

The probabilities for the univariate Normal distribution can be computed using nag_cumul_normal (s15abc) and nag_cumul_normal_complem (s15acc).

This example reads values of

x

and

y

for a bivariate Normal distribution along with the value of

ρ

and computes the lower tail probabilities.