/*************************************************************************
Binomial distribution
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
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialdistribution(int k, int n, double p)
{
double result;
double dk;
double dn;
ap::ap_error::make_assertion(ap::fp_greater_eq(p,0)&&ap::fp_less_eq(p,1), "Domain error in BinomialDistribution");
ap::ap_error::make_assertion(k>=-1&&k<=n, "Domain error in BinomialDistribution");
if( k==-1 )
{
result = 0;
return result;
}
if( k==n )
{
result = 1;
return result;
}
dn = n-k;
if( k==0 )
{
dk = pow(1.0-p, dn);
}
else
{
dk = k+1;
dk = incompletebeta(dn, dk, 1.0-p);
}
result = dk;
return result;
}
void MyEff(double n, double N, double& eff, double& errp, double& errn)
{
TF1* gaus = new TF1("gaus","TMath::Gaus(x,0,1,1)");
double lowerB = gaus->Integral(-1,0);
double upperB = 1- lowerB;
eff = n/N;
if(n > N)
{
errp = errn = -999;
return;
}
double a = n+1;
double b = N-n+1;
double integralPeak = incompletebeta(a, b, eff);
if(integralPeak >= lowerB && integralPeak <= upperB)
{
double xmin = inverseBeta(integralPeak - lowerB, a, b);
double xmax = inverseBeta(integralPeak + lowerB, a, b);
errp = xmax-eff;
errn = eff-xmin;
return;
}
else if(integralPeak < lowerB)
{
double xmax = inverseBeta(2*lowerB, a, b);
errp = xmax-eff;
errn = eff;
return;
}
else if(integralPeak > upperB)
{
double xmin = inverseBeta(1-2*lowerB, a, b);
errp = 1-eff;
errn = eff-xmin;
return;
}
else{
cout << "There is a bug !!!" << endl;
errp = errn = -999;
return;
}
}
double inverseBeta(double p, double alpha, double beta, double A, double B)
{
double x = 0;
double a = 0;
double b = 1;
A=0;
B=1;
double precision = DBL_EPSILON; // converge until there is 6 decimal places precision
while ((b - a) > precision)
{
x = (a + b) / 2;
if (incompletebeta(alpha, beta, x) > p)
{
b = x;
}
else
{
a = x;
}
}
if ((B > 0) && (A > 0))
{
x = x * (B - A) + A;
}
return x;
}
/*************************************************************************
Complemented binomial distribution
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
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialcdistribution(int k, int n, double p)
{
double result;
double dk;
double dn;
ap::ap_error::make_assertion(p>=0&&p<=1, "Domain error in BinomialDistributionC");
ap::ap_error::make_assertion(k>=-1&&k<=n, "Domain error in BinomialDistributionC");
if( k==-1 )
{
result = 1;
return result;
}
if( k==n )
{
result = 0;
return result;
}
dn = n-k;
if( k==0 )
{
if( p<0.01 )
{
dk = -expm1(dn*log1p(-p));
}
else
{
dk = 1.0-pow(1.0-p, dn);
}
}
else
{
dk = k+1;
dk = incompletebeta(dk, dn, p);
}
result = dk;
return result;
}
/*************************************************************************
Inverse binomial distribution
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
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invbinomialdistribution(int k, int n, double y)
{
double result;
double dk;
double dn;
double p;
ap::ap_error::make_assertion(p>=0&&p<=1, "Domain error in InvBinomialDistribution");
ap::ap_error::make_assertion(k>=0&&k<n, "Domain error in InvBinomialDistribution");
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 = incompletebeta(dn, dk, 0.5);
if( p>0.5 )
{
p = invincompletebeta(dk, dn, 1.0-y);
}
else
{
p = 1.0-invincompletebeta(dn, dk, y);
}
}
result = p;
return result;
}
/*************************************************************************
Student's t distribution
Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:
t
-
| |
- | 2 -(k+1)/2
| ( (k+1)/2 ) | ( x )
---------------------- | ( 1 + --- ) dx
- | ( k )
sqrt( k pi ) | ( k/2 ) |
| |
-
-inf.
Relation to incomplete beta integral:
1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
z = k/(k + t**2).
For t < -2, this is the method of computation. For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.
ACCURACY:
Tested at random 1 <= k <= 25. The "domain" refers to t.
Relative error:
arithmetic domain # trials peak rms
IEEE -100,-2 50000 5.9e-15 1.4e-15
IEEE -2,100 500000 2.7e-15 4.9e-17
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double studenttdistribution(int k, double t)
{
double result;
double x;
double rk;
double z;
double f;
double tz;
double p;
double xsqk;
int j;
ap::ap_error::make_assertion(k>0);
if( t==0 )
{
result = 0.5;
return result;
}
if( t<-2.0 )
{
rk = k;
z = rk/(rk+t*t);
result = 0.5*incompletebeta(0.5*rk, 0.5, z);
return result;
}
if( t<0 )
{
x = -t;
}
else
{
x = t;
}
rk = k;
z = 1.0+x*x/rk;
if( k%2!=0 )
{
xsqk = x/sqrt(rk);
p = atan(xsqk);
if( k>1 )
{
f = 1.0;
tz = 1.0;
j = 3;
while(j<=k-2&&tz/f>ap::machineepsilon)
{
tz = tz*((j-1)/(z*j));
f = f+tz;
j = j+2;
}
p = p+f*xsqk/z;
}
p = p*2.0/ap::pi();
}
else
{
f = 1.0;
tz = 1.0;
j = 2;
while(j<=k-2&&tz/f>ap::machineepsilon)
{
tz = tz*((j-1)/(z*j));
f = f+tz;
j = j+2;
}
p = f*x/sqrt(z*rk);
}
if( t<0 )
{
p = -p;
}
result = 0.5+0.5*p;
return result;
}
