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/* bdtrl.c
*
* Binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtrl();
*
* y = bdtrl( 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 (k,n,p) with a and b between 0
* and 10000 and p between 0 and 1.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,10000 3000 1.6e-14 2.2e-15
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtrl domain k < 0 0.0
* n < k
* x < 0, x > 1
*
*/
/* bdtrcl()
*
* Complemented binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtrcl();
*
* y = bdtrcl( 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:
*
* See incbet.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtrcl domain x<0, x>1, n<k 0.0
*/
/* bdtril()
*
* Inverse binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtril();
*
* p = bdtril( 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:
*
* See incbi.c.
* Tested at random k, n between 1 and 10000. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,1 3500 2.0e-15 8.2e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtril domain k < 0, n <= k 0.0
* x < 0, x > 1
*/
/* bdtr() */
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/
#include <math.h>
#ifdef ANSIPROT
extern long double incbetl ( long double, long double, long double );
extern long double incbil ( long double, long double, long double );
extern long double powl ( long double, long double );
extern long double expm1l ( long double );
extern long double log1pl ( long double );
#else
long double incbetl(), incbil(), powl(), expm1l(), log1pl();
#endif
long double bdtrcl( k, n, p )
int k, n;
long double p;
{
long double dk, dn;
if( (p < 0.0L) || (p > 1.0L) )
goto domerr;
if( k < 0 )
return( 1.0L );
if( n < k )
{
domerr:
mtherr( "bdtrcl", DOMAIN );
return( 0.0L );
}
if( k == n )
return( 0.0L );
dn = n - k;
if( k == 0 )
{
if( p < .01L )
dk = -expm1l( dn * log1pl(-p) );
else
dk = 1.0L - powl( 1.0L-p, dn );
}
else
{
dk = k + 1;
dk = incbetl( dk, dn, p );
}
return( dk );
}
long double bdtrl( k, n, p )
int k, n;
long double p;
{
long double dk, dn, q;
if( (p < 0.0L) || (p > 1.0L) )
goto domerr;
if( (k < 0) || (n < k) )
{
domerr:
mtherr( "bdtrl", DOMAIN );
return( 0.0L );
}
if( k == n )
return( 1.0L );
q = 1.0L - p;
dn = n - k;
if( k == 0 )
{
dk = powl( q, dn );
}
else
{
dk = k + 1;
dk = incbetl( dn, dk, q );
}
return( dk );
}
long double bdtril( k, n, y )
int k, n;
long double y;
{
long double dk, dn, p;
if( (y < 0.0L) || (y > 1.0L) )
goto domerr;
if( (k < 0) || (n <= k) )
{
domerr:
mtherr( "bdtril", DOMAIN );
return( 0.0L );
}
dn = n - k;
if( k == 0 )
{
if( y > 0.8L )
p = -expm1l( log1pl(y-1.0L) / dn );
else
p = 1.0L - powl( y, 1.0L/dn );
}
else
{
dk = k + 1;
p = incbetl( dn, dk, y );
if( p > 0.5 )
p = incbil( dk, dn, 1.0L-y );
else
p = 1.0 - incbil( dn, dk, y );
}
return( p );
}
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