This function evaluates the F cumulative distribution function.

Function Return Value

FDF — Function value, the probability that an F random variable takes a value less than or equal to the input F.   (Output)

Required Arguments

F — Argument for which the F cumulative distribution function is to be evaluated.   (Input)

DFN — Numerator degrees of freedom.   (Input)
DFN must be positive.

DFD — Denominator degrees of freedom.   (Input)
DFD must be positive.

Optional Arguments

COMPLEMENT — Logical. If .TRUE., the complement of the F cumulative distribution function is evaluated.  If .FALSE., the F cumulative distribution function is evaluated.   (Input)
See the Description section for further details on the use of COMPLEMENT.
Default: COMPLEMENT = .FALSE..

FORTRAN 90 Interface

Generic:                              FDF (F, DFN, DFD [,…])

Specific:                             The specific interface names are S_FDF and D_FDF.

FORTRAN 77 Interface

Single:                                FDF (F, DFN, DFD)

Double:                              The double precision name is DFDF.

Description

Function FDF evaluates the distribution function of a Snedecor's F random variable with DFN numerator degrees of freedom and DFD denominator degrees of freedom. The function is evaluated by making a transformation to a beta random variable and then using the routine BETDF. If X is an F variate with v1 and v2 degrees of freedom and Y = v1X/(v2 + v1X), then Y is a beta variate with parameters p = v1/2 and q = v2/2. The function FDF also uses a relationship between F random variables that can be expressed as follows.

FDF(X, DFN, DFD) = 1.0 − FDF(1.0/X, DFD, DFN)

If COMPLEMENT = .TRUE., the value of FDF at the point x is 1− p, where 1− p is the probability that the random variable takes a value greater than x. In those situations where the desired end result is 1− p, the user can achieve greater accuracy in the right tail region by using the result returned by FDF with the optional argument COMPLEMENT set to .TRUE. rather than by using
1− p where p is the result returned by FDF with COMPLEMENT set to .FALSE..

 

Figure 11- 10   F Distribution Function

Comments

Informational error

Type Code

1         3                  Since the input argument F is not positive, the distribution function is zero at F.

Example

In this example, we find the probability that an F random variable with one numerator and one denominator degree of freedom is greater than 648.

 

      USE UMACH_INT

      USE FDF_INT

      IMPLICIT   NONE

      INTEGER    NOUT

      REAL       DFD, DFN, F, P

!

      CALL UMACH (2, NOUT)

      F   = 648.0

      DFN = 1.0

      DFD = 1.0

      P   = FDF(F,DFN,DFD, COMPLEMENT=.TRUE.)

      WRITE (NOUT,99999) P

99999 FORMAT (' The probability that an F(1,1) variate is greater ', &

            'than 648 is ', F6.4)

      END

Output

 

The probability that an F(1, 1) variate is greater than 648 is 0.0250

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