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. \(0<a\leq 10^6\)

\(b\) (input, double)

the second shape parameter, b, of the required beta distribtution.\(0<b\leq 10^6\)

\(lambda\) (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.