This fuction evaluates the standard normal (Gaussian) cumulative distribution function.
Function Return Value
ANORDF — Function value, the probability that a normal random variable takes a value less than or equal to X. (Output)
Required Arguments
X — Argument for which the normal cumulative distribution function is to be evaluated. (Input)
FORTRAN 90 Interface
Generic: ANORDF (X)
Specific: The specific interface names are S_ANORDF and D_ANORDF.
FORTRAN 77 Interface
Single: ANORDF (X)
Double: The double precision name is DNORDF.
Description
Function ANORDF evaluates the cumulative distribution function, Φ, of a standard normal (Gaussian) random variable, that is,
The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
The standard normal distribution (for which ANORDF is the distribution function) has mean of 0 and variance of 1. The probability that a normal random variable with mean and variance σ2 is less than y is given by ANORDF evaluated at (y − μ)/σ.
Φ(x) is evaluated by use of the complementary error function, erfc. (See ERFC, IMSL MATH/LIBRARY Special Functions). The relationship is:
Figure 11- 6 Standard Normal Distribution Function
Example
Suppose X is a normal random variable with mean 100 and variance 225. In this example, we find the probability that X is less than 90, and the probability that X is between 105 and 110.
USE UMACH_INT
USE ANORDF_INT
IMPLICIT NONE
INTEGER NOUT
REAL P, X1, X2
!
CALL UMACH (2, NOUT)
X1 = (90.0-100.0)/15.0
P = ANORDF(X1)
WRITE (NOUT,99998) P
99998 FORMAT (' The probability that X is less than 90 is ', F6.4)
X1 = (105.0-100.0)/15.0
X2 = (110.0-100.0)/15.0
P = ANORDF(X2) - ANORDF(X1)
WRITE (NOUT,99999) P
99999 FORMAT (' The probability that X is between 105 and 110 is ', &
F6.4)
END
Output
The probability that X is less than 90 is 0.2525
The probability that X is between 105 and 110 is 0.1169
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