G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01HAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01HAF returns the lower tail probability for the bivariate Normal distribution, via the routine name.

## 2  Specification

 FUNCTION G01HAF ( X, Y, RHO, IFAIL)
 REAL (KIND=nag_wp) G01HAF
 INTEGER IFAIL REAL (KIND=nag_wp) X, Y, RHO

## 3  Description

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).

## 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  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: $x$, the first argument for which the bivariate Normal distribution function is to be evaluated.
2:     Y – REAL (KIND=nag_wp)Input
On entry: $y$, the second argument for which the bivariate Normal distribution function is to be evaluated.
3:     RHO – REAL (KIND=nag_wp)Input
On entry: $\rho$, the correlation coefficient.
Constraint: $-1.0\le {\mathbf{RHO}}\le 1.0$.
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{RHO}}<-1.0$, or ${\mathbf{RHO}}>1.0$.
If on exit ${\mathbf{IFAIL}}={\mathbf{1}}$ then G01HAF returns zero.

## 7  Accuracy

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

The probabilities for the univariate Normal distribution can be computed using S15ABF and S15ACF.

## 9  Example

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.

### 9.1  Program Text

Program Text (g01hafe.f90)

### 9.2  Program Data

Program Data (g01hafe.d)

### 9.3  Program Results

Program Results (g01hafe.r)