BINORMALCDF Function

Evaluates the bivariate normal distribution function.

Usage

result = BINORMALCDF(x, y, rho)

Input Parameters

x—The x-coordinate of the point for which the bivariate normal distribution function is to be evaluated.

y—The y-coordinate of the point for which the bivariate normal distribution function is to be evaluated.

rho—Correlation coefficient.

Returned Value

result—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.

Input Keywords

Double—If present and nonzero, double precision is used.

Discussion

Function BINORMALCDF evaluates the distribution function F of a bivariate normal distribution with means of zero, variances of 1, and correlation of rho; that is, r = rho and |r| < 1.

 

To determine the probability that U £ u₀ and V £ v₀, where (U, V) is a bivariate normal random variable with mean m = (m_U, m_V) and the following variance-covariance matrix:

 

transform (U, V)^T to a vector with zero means and unit variances. The input to BINORMALCDF would be as follows:

, , and

The BINORMALCDF function uses the method of Owen (1962, 1965). For |r| = 1, the distribution function is computed based on the univariate statistic Z = min(x, y) and on the normal distribution NORMALCDF.

Example

Suppose (x, y) is a bivariate normal random variable with mean (0, 0) and the following variance-covariance matrix:

 

This example finds the probability that x is less than –2.0 and y is less than 0.0.

; Define x, y, and rho.

x = -2 

y = 0

rho = .9

; Call BINORMALCDF and output the results.

p = BINORMALCDF(x, y, rho)

PM, 'P((x < -2.0) and (y < 0.0)) = ', p, Format = '(a29, f8.4)'

; PV-WAVE prints: P((x < -2.0) and (y < 0.0)) = 0.0228