The IMSL_FCDF function evaluates the F distribution function. Using a keyword, the inverse of the F distribution function can be evaluated.

The IMSL_FCDF function evaluates the distribution function of a Snedecor’s F random variable with dfnum and dfden. The function is evaluated by making a transformation to a beta random variable and then evaluating the incomplete beta function. If X is an F variate with ν1 and ν2 degrees of freedom and Y = (ν1X)/(ν2 + ν1X), then Y is a beta variate with parameters p = ν1/2 and q = ν2/2. The IMSL_FCDF function also uses a relationship between F random variables that is expressed as follows: FF(f, ν1, ν2) = 1 – FF(1/f, ν2, ν1), where FF is the distribution function for an F random variable.

If INVERSE is specified, the IMSL_FCDF function evaluates the inverse distribution function of a Snedecor’s F random variable with ν1 = dfnum numerator degrees of freedom and ν2 = dfden denominator degrees of freedom. The function is evaluated by making a transformation to a beta random variable and then evaluating the inverse of an incomplete beta function.

## Example

This example finds the probability that an F random variable with one numerator and one denominator degree of freedom is greater than 648.

`f = 648`
`p = 1 - IMSL_FCDF(f, 1, 1)`
`PM, p, Title = 'The probability that an F(1,1) ' + \$`
`  'variate is greater than 648 is:'`
` `
`The probability that an F(1,1) variate is greater than 648 is:`
`0.0249959`

## Syntax

Result = IMSL_FCDF(F, Dfnum, Dfden [, /DOUBLE] [, /INVERSE])

## Return Value

The probability that an F random variable takes a value less than or equal to the input point f.

## Arguments

### Dfden

Denominator degrees of freedom. Parameter Dfden must be positive.

### Dfnum

Numerator degrees of freedom. Parameter Dfnum must be positive.

### F

Expression for which the F distribution function is to be evaluated.

## Keywords

### DOUBLE (optional)

If present and nonzero, double precision is used.

### INVERSE (optional)

If present and nonzero, evaluates the inverse of the F distribution function. If inverse is specified, argument f represents the probability for which the inverse of the F distribution function is to be evaluated. In this case, f must be in the open interval (0.0, 1.0).

## Errors

### Fatal Errors

STAT_F_INVERSE_OVERFLOW: IMSL_FCDF is set to machine infinity since overflow would occur upon modifying the inverse value for the F distribution with the result obtained from the inverse beta distribution.

## Version History

 6.4 Introduced