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bdtr.c
/*                                        bdtr.c
 *
 *    Binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtr();
 *
 * y = bdtr( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms 0 through k of the Binomial
 * probability density:
 *
 *   k
 *   --  ( n )   j      n-j
 *   >   (   )  p  (1-p)
 *   --  ( j )
 *  j=0
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtr( k, n, p ) = incbet( n-k, k+1, 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
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      4.3e-15     2.6e-16
 * See also incbet.c.
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtr domain         k < 0            0.0
 *                     n < k
 *                     x < 0, x > 1
 */
/*                                       bdtrc()
 *
 *    Complemented binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtrc();
 *
 * y = bdtrc( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 through n of the Binomial
 * probability density:
 *
 *   n
 *   --  ( n )   j      n-j
 *   >   (   )  p  (1-p)
 *   --  ( j )
 *  j=k+1
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p).
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      6.7e-15     8.2e-16
 *  For p between 0 and .001:
 *    IEEE     0,100       100000      1.5e-13     2.7e-15
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtrc domain      x<0, x>1, n<k       0.0
 */
/*                                       bdtri()
 *
 *    Inverse binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtri();
 *
 * p = bdtr( k, n, y );
 *
 * DESCRIPTION:
 *
 * Finds the event probability p such that the sum of the
 * terms 0 through k of the Binomial probability density
 * is equal to the given cumulative probability y.
 *
 * This is accomplished using the inverse beta integral
 * function and the relation
 *
 * 1 - p = incbi( n-k, k+1, y ).
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p).
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      2.3e-14     6.4e-16
 *    IEEE     0,10000     100000      6.6e-12     1.2e-13
 *  For p between 10^-6 and 0.001:
 *    IEEE     0,100       100000      2.0e-12     1.3e-14
 *    IEEE     0,10000     100000      1.5e-12     3.2e-14
 * See also incbi.c.
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtri domain     k < 0, n <= k         0.0
 *                  x < 0, x > 1
 */

/*                                              bdtr() */


/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

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

    if (p < 0.0 || p > 1.0)
      goto domerr;
    if (k < 0)
      return 1.0;

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

    if (k == n)
      return 0.0;
    dn = n - k;
    if (k == 0) {
      if (p < .01)
          dk = -cephes_exp(dn * cephes_log(-p));
      else
          dk = 1.0 - pow(1.0 - p, dn);
    } else {
      dk = k + 1;
      dk = incbet(dk, dn, p);
    }

    return dk;
}

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

    if (p < 0.0 || p > 1.0)
      goto domerr;
    if (k < 0 || n < k) {
    domerr:
      mtherr("bdtr", CEPHES_DOMAIN);
      return 0.0;
    }

    if (k == n)
      return 1.0;

    dn = n - k;
    if (k == 0) {
      dk = pow(1.0 - p, dn);
    } else {
      dk = k + 1;
      dk = incbet(dn, dk, 1.0 - p);
    }

    return dk;
}

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

    if (y < 0.0 || y > 1.0)
      goto domerr;
    if (k < 0 || n <= k) {
    domerr:
      mtherr("bdtri", CEPHES_DOMAIN);
      return 0.0;
    }

    dn = n - k;
    if (k == 0) {
      if (y > 0.8)
          p = -cephes_exp(cephes_log(y - 1.0) / dn);
      else
          p = 1.0 - pow(y, 1.0 / dn);
    } else {
      dk = k + 1;
      p = incbet(dn, dk, 0.5);
      if (p > 0.5)
          p = incbi(dk, dn, 1.0 - y);
      else
          p = 1.0 - incbi(dn, dk, y);
    }

    return p;
}

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