g01ha returns the lower tail probability for the bivariate Normal distribution.

Syntax

C#
public static double g01ha( double x, double y, double rho, out int ifail )

Visual Basic
Public Shared Function g01ha ( _ x As Double, _ y As Double, _ rho As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ha( double x, double y, double rho, [OutAttribute] int% ifail )

F#
static member g01ha : x : float * y : float * rho : float * ifail : int byref -> float

Parameters

x: Type: System..::..Double
On entry: $x$ , the first argument for which the bivariate Normal distribution function is to be evaluated.

y: Type: System..::..Double
On entry: $y$ , the second argument for which the bivariate Normal distribution function is to be evaluated.

rho: Type: System..::..Double
On entry: $ρ$ , the correlation coefficient.

Constraint: $- 1.0 \leq rho \leq 1.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01ha returns the lower tail probability for the bivariate Normal distribution.

Description

For the two random variables

(X, Y)

following a bivariate Normal distribution with

E [X] = 0, E [Y] = 0, E [X^{2}] = 1, E [Y^{2}] = 1 and E [X Y] = ρ,

the lower tail probability is defined by:

P (X \leq x, Y \leq y : ρ) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} \int_{- \infty}^{y} \int_{- \infty}^{x} \exp (- \frac{(X^{2} - 2 ρ X Y + Y^{2})}{2 (1 - ρ^{2})}) d X d Y .

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).

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$rho < - 1.0$ ,
or	$rho > 1.0$ .

If on exit

ifail = 1

then g01ha returns zero.

$ifail = -9000$: An error occured, see message report.

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 \times 10^{- 16}

Parallelism and Performance

None.

Further Comments

The probabilities for the univariate Normal distribution can be computed using s15ab and s15ac.

Example

This example reads values of

x

and

y

for a bivariate Normal distribution along with the value of

ρ

and computes the lower tail probabilities.

Example program (C#): g01hae.cs

Example program data: g01hae.d

Example program results: g01hae.r