# NAG FL Interfaceg01haf (prob_​bivariate_​normal)

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## 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 <nag.h>
 double g01haf_ (const double *x, const double *y, const double *rho, Integer *ifail)
The routine may be called by the names g01haf or nagf_stat_prob_bivariate_normal.

## 3Description

For the two random variables $\left(X,Y\right)$ following a bivariate Normal distribution with
 $E[X]=0, E[Y]=0, E[X2]=1, E[Y2]=1 and E[XY]=ρ,$
the lower tail probability is defined by:
 $P(X≤x,Y≤y:ρ)=12π⁢1-ρ2 ∫-∞y ∫-∞x exp(- (X2-2⁢ρXY+Y2) 2⁢(1-ρ2) ) dXdY.$
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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)