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.

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Cumulative Distribution Functions (CDF)

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