# 3.5.3.1.17 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.
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.