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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 );
+}