# NAG C Library Function Document

## 1Purpose

nag_bivariate_normal_dist (g01hac) returns the lower tail probability for the bivariate Normal distribution.

## 2Specification

 #include #include
 double nag_bivariate_normal_dist (double x, double y, double rho, NagError *fail)

## 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:    $\mathbf{x}$doubleInput
On entry: $x$, the first argument for which the bivariate Normal distribution function is to be evaluated.
2:    $\mathbf{y}$doubleInput
On entry: $y$, the second argument for which the bivariate Normal distribution function is to be evaluated.
3:    $\mathbf{rho}$doubleInput
On entry: $\rho$, the correlation coefficient.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

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.
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, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rho}}\le 1.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rho}}\ge -1.0$.

## 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

nag_bivariate_normal_dist (g01hac) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

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

## 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 (g01hace.c)

### 10.2Program Data

Program Data (g01hace.d)

### 10.3Program Results

Program Results (g01hace.r)