BNRDF

This function evaluates the bivariate normal cumulative distribution function.

Function Return Value

BNRDF — Function 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. (Output)

Required Arguments

X — One argument for which the bivariate normal distribution function is to be evaluated. (Input)

Y — The other argument for which the bivariate normal distribution function is to be evaluated. (Input)

RHO — Correlation coefficient. (Input)

FORTRAN 90 Interface

Generic: BNRDF (X, Y, RHO)

Specific: The specific interface names are S_BNRDF and D_BNRDF.

FORTRAN 77 Interface

Single: BNRDF (X, Y, RHO)

Double: The double precision name is DBNRDF.

Description

Function BNRDF evaluates the cumulative 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 BNRDF would be X = (u0 ‑ μU)/σU, Y = (v0 ‑ μV)/σV, and ρ = σUV/(σUσV).

Function BNRDF 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 ANORDF.

Example

Suppose (X, Y) is a bivariate normal random variable with mean (0, 0) and variance‑covariance matrix

 

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

 

USE BNRDF_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER NOUT

REAL P, RHO, X, Y

!

CALL UMACH (2, NOUT)

X = -2.0

Y = 0.0

RHO = 0.9

P = BNRDF(X,Y,RHO)

WRITE (NOUT,99999) P

99999 FORMAT (' The probability that X is less than -2.0 and Y ', &

'is less than 0.0 is ', F6.4)

END

Output

 

The probability that X is less than –2.0 and Y is less than 0.0 is 0.0228