# Ncbetacdf

## Definition:

$prob = ncbetacdf(x, a, b, lambda)$ computes the cdf with the lower tail of the non-central beta distribution. The lower tail probability for the non-central beta distribution with shape parameters $a$ and $b$ and non-centrality parameter $\lambda$ ,$p(B\leq \beta )$ is defined by:

$P(B\leq \beta )=\sum_{j=0}^\infty e^{-\lambda /2}\frac{(\lambda /2)^j}{j!}P_b(B\leq \beta )$

where

$P_b(B\leq \beta )=\frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)}\int_0^\beta B^{a-1}(1-B)^{b-1}dB$

which is the central beta probability function or incomplete beta function.

## Parameters:

$x$ (intput, double)
the deviate,$\beta$ ,from the beta distribution, for which $P(B\leq \beta )$ , is to be found.$0\leq x\leq 1.0$
$a$ (input, double)
the first shape parameter, a, of the required beta distribtution. $0
$b$ (input, double)
the second shape parameter, b, of the required beta distribtution.$0
$lambda$ (input, double)
the non-centrality parameter,$\lambda$ , of the required beta distribution,$0\leq lambda \leq -2.0\times \log (U)$ , where $U$ is the safe range parameters as defined by NAG nag_real_safe_small_number (X02AMC). See chapter X02 in the NAG documentation.
$prob$ (output,double)
the probability.