3.5.3.1.20 Ncfcdf

Definition:

\[prob = ncfcdf(f, df1, df2, lambda)\] computes the probability associated with the lower tail of the non-central \(\digamma\) or variance-ratio distribution.

The lower tail probability of the non-central F-distribution with \(\nu _1\) and \(\nu _2\) degrees of freedom and non-centrality parameter \(\lambda\), \(P(\digamma \leq f)\)is defined by:
\(P(\digamma \leq f)\)=\(\int_\lambda ^f P(\digamma )d\digamma\),

Where

\(P(\digamma )=\sum_{j=0}^\infty e^{-\lambda /2}\frac{(\lambda /2)^j}{j!}\times\frac{(\nu _1+2j)^{(\nu _1+2j)/2}\nu _2^{\nu _2/2}}{B((\nu _1+2j)/2,\nu _2/2)}\times\) \( u^{(\nu _1+2j-2)/2}\left[ \nu _2+(\nu _1+2j)u\right] ^{-(\nu _1+2j+\nu _2)/2} \)

and \(B\left( \cdot ,\cdot \right)\) is the beta function.

Parameters:

f (input,double)

The deviate from the non-central F-distribution,. \(f{>}0\).

df1 (input,double)

The degrees of freedom of the numerator variance,\(\nu _1\). \(0<df1\leq 1.0e6\).

df2 (input,double)

The degrees of freedom of the denominator variance,\(\nu _2\). \(df2>0.\)

lambda (input,double)

The non-centrality parameter,\(lambda\), of the required beta distribution,\(0\leq lambda\leq -2.0\times \log \left( U\right) ,\)where U is the safe range parameters as defined by by NAG nag_real_safe_small_number (X02AMC). See chapter X02 in the NAG documentation.

prob (output,double)

The probability.