bivariate_normal_cdf

Evaluates the bivariate normal distribution function.

Synopsis

#include <imsls.h>

float imsls_f_bivariate_normal_cdf (float x, float y, float rho)

The type double function is imsls_d_bivariate_normal_cdf.

Required Arguments

float x (Input)
The x-coordinate of the point for which the bivariate normal distribution function is to be evaluated.

float y (Input)
The y-coordinate of the point for which the bivariate normal distribution function is to be evaluated.

float rho (Input)
Correlation coefficient.

Return Value

The probability that a bivariate normal random variable with correlation rho takes a value less than or equal to x and less than or equal to y.

Description

Function imsls_f_bivariate_normal_cdf evaluates the distribution function F of a bivariate normal distribution with means of zero, variances of one, and correlation of rho; that is, with ρ = rho, and ∣ρ∣ < 1,

 

To determine the probability that U ≤ u0 and V ≤ v0, where (U, V)T is a bivariate normal random variable with mean μ = (μU, μV)T and variance-covariance matrix

 

transform (U, V)T to a vector with zero means and unit variances. The input to imsls_f_bivariate_normal_cdf would be X = (u0 – μU)/σU, Y = (v0 – μV)/σV, and ρ = σUV/(σUσV).

Function imsls_f_bivariate_normal_cdf uses the method of Owen (1962, 1965). Computation of Owen’s T-function is based on code by M. Patefield and D. Tandy (2000). For ∣ρ∣ = 1, the distribution function is computed based on the univariate statistic, Z = min(x, y), and on the normal distribution function imsls_f_normal_cdf.

Example

Suppose (X, Y) is a bivariate normal random variable with mean (0, 0) and variance-covariance matrix as follows:

 

In this example, we find the probability that Xis less than −2.0 and Y is less than 0.0.

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

    float  rho = 0.9, x = -2.0, y = 0.0, p;

 

    p = imsls_f_bivariate_normal_cdf(x, y, rho);

    printf(" The probability that X is less than %4.1f\n"

        " and Y is less than %3.1f is %6.4f\n", x, y, p);

}

 

Output

 

The probability that X is less than -2.0

and Y is less than 0.0 is 0.0228