IMSL_BINORMALCDF

Syntax | Return Value | Arguments | Keywords | Discussion | Example | Version History

The IMSL_BINORMALCDF function evaluates the bivariate normal distribution function.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = IMSL_BINORMALCDF(x, y, rho [, /DOUBLE])

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.

Arguments

rho

Correlation coefficient.

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.

Keywords

DOUBLE

If present and nonzero, double precision is used.

Discussion

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

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

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

,

,

and

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

x = -2  
y = 0 
rho = .9 
; Define x, y, and rho. 
p = IMSL_BINORMALCDF(x, y, rho) 
; Call IMSL_BINORMALCDF and output the results. 
PM, 'P((x < -2.0) and (y < 0.0)) = ', p, FORMAT = '(a29, f8.4)' 
 
P((x < -2.0) and (y < 0.0)) = 0.0228 

Version History