# NAG Library Routine Document

## 1Purpose

g01haf returns the lower tail probability for the bivariate Normal distribution.

## 2Specification

Fortran Interface
 Function g01haf ( x, y, rho,
 Real (Kind=nag_wp) :: g01haf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, y, rho
#include <nagmk26.h>
 double g01haf_ (const double *x, const double *y, const double *rho, Integer *ifail)

## 3Description

For the two random variables $\left(X,Y\right)$ following a bivariate Normal distribution with
 $EX=0, EY=0, EX2=1, EY2=1 and EXY=ρ,$
the lower tail probability is defined by:
 $PX≤x,Y≤y:ρ=12π⁢1-ρ2 ∫-∞y ∫-∞x exp- X2- 2ρ XY+Y2 21-ρ2 dXdY.$
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).

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the first argument for which the bivariate Normal distribution function is to be evaluated.
2:     $\mathbf{y}$ – Real (Kind=nag_wp)Input
On entry: $y$, the second argument for which the bivariate Normal distribution function is to be evaluated.
3:     $\mathbf{rho}$ – Real (Kind=nag_wp)Input
On entry: $\rho$, the correlation coefficient.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rho}}\ge -1.0$.
On entry, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rho}}\le 1.0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Accuracy of the hybrid algorithm implemented here is discussed in Genz (2004). This algorithm should give a maximum absolute error of less than $5×{10}^{-16}$.

## 8Parallelism and Performance

g01haf is not threaded in any implementation.

The probabilities for the univariate Normal distribution can be computed using s15abf and s15acf.

## 10Example

This example reads values of $x$ and $y$ for a bivariate Normal distribution along with the value of $\rho$ and computes the lower tail probabilities.

### 10.1Program Text

Program Text (g01hafe.f90)

### 10.2Program Data

Program Data (g01hafe.d)

### 10.3Program Results

Program Results (g01hafe.r)