naginterfaces.library.stat.prob_​bivariate_​normal

naginterfaces.library.stat.prob_bivariate_normal(x, y, rho)

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

For full information please refer to the NAG Library document for g01ha

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01haf.html

Parameters

xfloat

$x$, the first argument for which the bivariate Normal distribution function is to be evaluated.

yfloat

$y$, the second argument for which the bivariate Normal distribution function is to be evaluated.

rhofloat

$\rho$, the correlation coefficient.

Returns

pfloat

The lower tail probability for the bivariate Normal distribution.

Raises

NagValueError

(errno $1$)

On entry, $rho=⟨value⟩$.

Constraint: $rho\le 1.0$.

(errno $1$)

On entry, $rho=⟨value⟩$.

Constraint: $rho\ge -1.0$.

Notes

For the two random variables $(X,Y)$ following a bivariate Normal distribution with

$$E\left[X\right]=0\text{,}E\left[Y\right]=0\text{,}E\left[X^2\right]=1\text{,}E\left[Y^2\right]=1\text{and}E\left[XY\right]=\rho \text{,}$$

the lower tail probability is defined by:

$$P(X\le x,Y\le y:\rho )=\frac{1}{2\pi \sqrt{1-{\rho }^2}}{\int }_{-\infty }^y{\int }_{-\infty }^{x}exp(-\frac{(X^2-2\rho XY+Y^2)}{2(1-{\rho }^2)})dXdY\text{.}$$

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