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 ≤ ν0, 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 , , and .
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
