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cumF(x,df1,df2 [,lam] [,upper:T or lower:F]), x, df1, df2 and lam REAL,
  elements of df1 and df2 >i 0 and lam >= 0

Keywords: probabilities
cumF(Val,df1,df2) computes the probabilities that an F random variable
with df1 and df2 degrees of freedom would be less than the elements of
the vector, matrix, or array Val.

cumF(Val,df1,df2,upper:T) and cumF(Val,df1,df2,lower:F) compute upper
tail probabilities.  For large Val, the result may preserve significant
digits that are lost when computing the upper tail probability by 1 -

cumF(Val,df1,df2,lam [,upper:T or lower:F]) computes similar
probabilities for non-central F with noncentrality parameter lam.

Any of Val, df1, df2, or lam that are not scalars (single numbers) must
be vectors, matrices, or arrays with the same size and shape which will
also be the size and shape of the result.

The degrees of freedom must be positive REAL numbers (not necessarily
integers).  Upper tail areas of F can be computed as 1 - cumF().

Compute P-value for F-statistic following anova():
  Cmd> cumF((SS[2]/DF[2])/(SS[5]/DF[5]), DF[2], DF[5], upper:T)

Compute 2-tail P-value for test of sigma_1 = sigma_2 based on sample
standard deviations s1 and s2 from independent normal samples.

  Cmd> 2*min(cumF(s1^2/s2^2, n1-1, n2-1), \
           cumF(s1^2/s2^2, n1-1, n2-1, upper:T)) # two tail P-value

See also invF(), cumbeta(), invbeta().

Gary Oehlert 2003-01-15