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nbdtr.c
/*                                        nbdtr.c
 *
 *    Negative binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, nbdtr();
 *
 * y = nbdtr( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms 0 through k of the negative
 * binomial distribution:
 *
 *   k
 *   --  ( n+j-1 )   n      j
 *   >   (       )  p  (1-p)
 *   --  (   j   )
 *  j=0
 *
 * In a sequence of Bernoulli trials, this is the probability
 * that k or fewer failures precede the nth success.
 *
 * The terms are not computed individually; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p), with p between 0 and 1.
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *    IEEE     0,100       100000      1.7e-13     8.8e-15
 * See also incbet.c.
 *
 */
/*                                       nbdtr.c
 *
 *    Complemented negative binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, nbdtrc();
 *
 * y = nbdtrc( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 to infinity of the negative
 * binomial distribution:
 *
 *   inf
 *   --  ( n+j-1 )   n      j
 *   >   (       )  p  (1-p)
 *   --  (   j   )
 *  j=k+1
 *
 * The terms are not computed individually; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p), with p between 0 and 1.
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *    IEEE     0,100       100000      1.7e-13     8.8e-15
 * See also incbet.c.
 */
/*                                       nbdtr.c
 *
 *    Functional inverse of negative binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, nbdtri();
 *
 * p = nbdtri( k, n, y );
 *
 * DESCRIPTION:
 *
 * Finds the argument p such that nbdtr(k,n,p) is equal to y.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,y), with y between 0 and 1.
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *    IEEE     0,100       100000      1.5e-14     8.5e-16
 * See also incbi.c.
 */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

double nbdtrc (int k, int n, double p)
{
    double dk, dn;

    if ((p < 0.0) || (p > 1.0)) {
      goto domerr;
    }

    if (k < 0) {
    domerr:
      mtherr("nbdtr", CEPHES_DOMAIN);
      return 0.0;
    }

    dk = k+1;
    dn = n;
    return incbet(dk, dn, 1.0 - p);
}

double nbdtr( int k, int n, double p )
{
    double dk, dn;

    if ((p < 0.0) || (p > 1.0))
      goto domerr;

    if (k < 0) {
    domerr:
      mtherr("nbdtr", CEPHES_DOMAIN);
      return 0.0;
    }

    dk = k+1;
    dn = n;
    return incbet(dn, dk, p);
}

double nbdtri( int k, int n, double p )
{
    double dk, dn, w;

    if ((p < 0.0) || (p > 1.0))
      goto domerr;

    if (k < 0) {
    domerr:
      mtherr("nbdtri", CEPHES_DOMAIN);
      return 0.0;
    }

    dk = k+1;
    dn = n;
    w = incbi(dn, dk, p);

    return w;
}

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