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### IMSL_BINORMALCDF

IMSL_BINORMALCDF

The IMSL_BINORMALCDF function evaluates the bivariate normal distribution function.

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 = (µU, µV) 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.

## Examples

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)'`

IDL prints:

`P((x < -2.0) and (y < 0.0)) = 0.0228`

## 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 (optional)

If present and nonzero, double precision is used.

## Version History

 6.4 Introduced

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