g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_bivariate_normal_dist (g01hac)

## 1  Purpose

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

## 2  Specification

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

## 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  Arguments

1:     xdoubleInput
On entry: $x$, the first argument for which the bivariate Normal distribution function is to be evaluated.
2:     ydoubleInput
On entry: $y$, the second argument for which the bivariate Normal distribution function is to be evaluated.
3:     rhodoubleInput
On entry: $\rho$, the correlation coefficient.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On any of the error conditions listed below nag_bivariate_normal_dist (g01hac) returns $0.0$.
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.
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: ${\mathbf{rho}}\ge -1.0$.

## 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 $5×{10}^{-16}$.

## 8  Parallelism and Performance

Not applicable.

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

## 10  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.

### 10.1  Program Text

Program Text (g01hace.c)

### 10.2  Program Data

Program Data (g01hace.d)

### 10.3  Program Results

Program Results (g01hace.r)