NON_CENTRAL_F_CDF Function

Evaluates the noncentral F cumulative distribution function (CDF).

Usage

result = NON_CENTRAL_F_CDF (f, df_numerator, df_denominator, lambda)

Input Parameters

f—Scalar float value for which the noncentral F cumulative distribution function is to be evaluated. f must be non-negative.

df_numerator—Scalar float value indicating the numerator degrees of freedom of the noncentral F distribution. df_numerator must be positive.

df_denominator—Scalar float value indicating the denominator degrees of freedom of the noncentral F distribution. df_denominator must be positive.

lambda—Scalar float value indicating the noncentrality parameter. lambda must be non-negative.

Returned Value

result—The probability that a noncentral F random variable takes a value less than or equal to f.

Input Keywords

Inverse—If present and nonzero, evaluates the inverse of the noncentral F cumulative distribution function. If inverse is specified, then f represents the probability for which the inverse of the noncentral F function is to be evaluated. f must be non-negative and less than one.

Double—If present and nonzero, then double precision is used.

Discussion

If X is a noncentral chi-square random variable with noncentrality parameter l and n₁ degrees of freedom, and Y is a chi-square random variable with n₂ degrees of freedom which is statistically independent of X, then:

F = (X/n₁)/(Y/n₂)

is a noncentral F-distributed random variable whose CDF is given by:

 

where the probability density function PDF(x) is given by:

 

 

 

 

and G(.) is the gamma function, n₁ = df_numerator, n₂ = df_denominator, l = lambda, and f = f.

With a noncentrality parameter of zero, the noncentral F distribution is the same as the F distribution.

Example

This example traces out a portion of a noncentral F cumulative distribution function with parameters df_numerator = 100, df_denominator = 10, and lambda = 10.

PRO t_non_central_f_cdf

   f = [0., .4, .8, 1.2, 1.6, 2.0, 2.8, 4.0] 

   df_numerator = 100.

   df_denominator = 10.

   lambda =10. 

   PRINT,"df_numerator:",$

          STRING(df_numerator,Format="(f4.0)") 

   PRINT,"df_denominator:",$

          STRING(df_denominator,Format="(f4.0)")  

   PRINT,"lambda:",$

          STRING(lambda,Format="(f4.0)") 

   PRINT,"    f       cdf(f)"

   FOR i=0L, 7 DO BEGIN  

      cdfv = NON_CENTRAL_F_CDF( f(i),df_numerator, $  

                           df_denominator, lambda) 

      PRINT,STRING(f(i),Format="(f5.1)"),"   ",$

            STRING(cdfv,Format="(E12.4)")  

   ENDFOR

END 

Output

df_numerator:    100

df_denominator:   10

lambda:           10

    f       cdf(f)

   0.0   0.0000e+000

   0.4   4.8879e-003

   0.8   2.0263e-001

   1.2   5.2114e-001

   1.6   7.3385e-001

   2.0   8.5041e-001

   2.8   9.4713e-001

   4.0   9.8536e-001