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,


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.


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.