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/* 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 <math.h>
#ifdef ANSIPROT
extern double incbet ( double, double, double );
extern double incbi ( double, double, double );
extern double pow ( double, double );
extern double log1p ( double );
extern double expm1 ( double );
#else
double incbet(), incbi(), pow(), log1p(), expm1();
#endif
double bdtrc( k, n, p )
int k, 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", DOMAIN );
return( 0.0 );
}
if( k == n )
return( 0.0 );
dn = n - k;
if( k == 0 )
{
if( p < .01 )
dk = -expm1( dn * log1p(-p) );
else
dk = 1.0 - pow( 1.0-p, dn );
}
else
{
dk = k + 1;
dk = incbet( dk, dn, p );
}
return( dk );
}
double bdtr( k, n, p )
int k, n;
double p;
{
double dk, dn;
if( (p < 0.0) || (p > 1.0) )
goto domerr;
if( (k < 0) || (n < k) )
{
domerr:
mtherr( "bdtr", 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( k, n, y )
int k, n;
double y;
{
double dk, dn, p;
if( (y < 0.0) || (y > 1.0) )
goto domerr;
if( (k < 0) || (n <= k) )
{
domerr:
mtherr( "bdtri", DOMAIN );
return( 0.0 );
}
dn = n - k;
if( k == 0 )
{
if( y > 0.8 )
p = -expm1( log1p(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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