# Boost C++ Libraries

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### boost/math/distributions/binomial.hpp

```// boost\math\distributions\binomial.hpp

// Copyright Paul A. Bristow 2007.

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.

// http://en.wikipedia.org/wiki/binomial_distribution

// Binomial distribution is the discrete probability distribution of
// the number (k) of successes, in a sequence of
// n independent (yes or no, success or failure) Bernoulli trials.

// It expresses the probability of a number of events occurring in a fixed time
// if these events occur with a known average rate (probability of success),
// and are independent of the time since the last event.

// The number of cars that pass through a certain point on a road during a given period of time.
// The number of spelling mistakes a secretary makes while typing a single page.
// The number of phone calls at a call center per minute.
// The number of times a web server is accessed per minute.
// The number of light bulbs that burn out in a certain amount of time.

// http://en.wikipedia.org/wiki/binomial_distribution

// Given a sample of N measured values k[i],
// we wish to estimate the value of the parameter x (mean)
// of the binomial population from which the sample was drawn.
// To calculate the maximum likelihood value = 1/N sum i = 1 to N of k[i]

// Also may want a function for EXACTLY k.

// And probability that there are EXACTLY k occurrences is
// exp(-x) * pow(x, k) / factorial(k)
// where x is expected occurrences (mean) during the given interval.
// For example, if events occur, on average, every 4 min,
// and we are interested in number of events occurring in 10 min,
// then x = 10/4 = 2.5

// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm

// The binomial distribution is used when there are
// exactly two mutually exclusive outcomes of a trial.
// These outcomes are appropriately labeled "success" and "failure".
// The binomial distribution is used to obtain
// the probability of observing x successes in N trials,
// with the probability of success on a single trial denoted by p.
// The binomial distribution assumes that p is fixed for all trials.

// P(x, p, n) = n!/(x! * (n-x)!) * p^x * (1-p)^(n-x)

// http://mathworld.wolfram.com/BinomialCoefficient.html

// The binomial coefficient (n; k) is the number of ways of picking
// k unordered outcomes from n possibilities,
// also known as a combination or combinatorial number.
// The symbols _nC_k and (n; k) are used to denote a binomial coefficient,
// and are sometimes read as "n choose k."
// (n; k) therefore gives the number of k-subsets  possible out of a set of n distinct items.

// For example:
//  The 2-subsets of {1,2,3,4} are the six pairs {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}, so (4; 2)==6.

// http://functions.wolfram.com/GammaBetaErf/Binomial/ for evaluation.

// But note that the binomial distribution
// (like others including the poisson, negative binomial & Bernoulli)
// is strictly defined as a discrete function: only integral values of k are envisaged.
// However because of the method of calculation using a continuous gamma function,
// it is convenient to treat it as if a continuous function,
// and permit non-integral values of k.
// To enforce the strict mathematical model, users should use floor or ceil functions
// on k outside this function to ensure that k is integral.

#ifndef BOOST_MATH_SPECIAL_BINOMIAL_HPP
#define BOOST_MATH_SPECIAL_BINOMIAL_HPP

#include <boost/math/distributions/fwd.hpp>
#include <boost/math/special_functions/beta.hpp> // for incomplete beta.
#include <boost/math/distributions/complement.hpp> // complements
#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
#include <boost/math/distributions/detail/inv_discrete_quantile.hpp> // error checks
#include <boost/math/special_functions/fpclassify.hpp> // isnan.
#include <boost/math/tools/roots.hpp> // for root finding.

#include <utility>

namespace boost
{
namespace math
{

template <class RealType, class Policy>
class binomial_distribution;

namespace binomial_detail{
// common error checking routines for binomial distribution functions:
template <class RealType, class Policy>
inline bool check_N(const char* function, const RealType& N, RealType* result, const Policy& pol)
{
if((N < 0) || !(boost::math::isfinite)(N))
{
*result = policies::raise_domain_error<RealType>(
function,
"Number of Trials argument is %1%, but must be >= 0 !", N, pol);
return false;
}
return true;
}
template <class RealType, class Policy>
inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)
{
if((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
{
*result = policies::raise_domain_error<RealType>(
function,
"Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol);
return false;
}
return true;
}
template <class RealType, class Policy>
inline bool check_dist(const char* function, const RealType& N, const RealType& p, RealType* result, const Policy& pol)
{
return check_success_fraction(
function, p, result, pol)
&& check_N(
function, N, result, pol);
}
template <class RealType, class Policy>
inline bool check_dist_and_k(const char* function, const RealType& N, const RealType& p, RealType k, RealType* result, const Policy& pol)
{
if(check_dist(function, N, p, result, pol) == false)
return false;
if((k < 0) || !(boost::math::isfinite)(k))
{
*result = policies::raise_domain_error<RealType>(
function,
"Number of Successes argument is %1%, but must be >= 0 !", k, pol);
return false;
}
if(k > N)
{
*result = policies::raise_domain_error<RealType>(
function,
"Number of Successes argument is %1%, but must be <= Number of Trials !", k, pol);
return false;
}
return true;
}
template <class RealType, class Policy>
inline bool check_dist_and_prob(const char* function, const RealType& N, RealType p, RealType prob, RealType* result, const Policy& pol)
{
if((check_dist(function, N, p, result, pol) && detail::check_probability(function, prob, result, pol)) == false)
return false;
return true;
}

template <class T, class Policy>
T inverse_binomial_cornish_fisher(T n, T sf, T p, T q, const Policy& pol)
{
BOOST_MATH_STD_USING
// mean:
T m = n * sf;
// standard deviation:
T sigma = sqrt(n * sf * (1 - sf));
// skewness
T sk = (1 - 2 * sf) / sigma;
// kurtosis:
// T k = (1 - 6 * sf * (1 - sf) ) / (n * sf * (1 - sf));
// Get the inverse of a std normal distribution:
T x = boost::math::erfc_inv(p > q ? 2 * q : 2 * p, pol) * constants::root_two<T>();
// Set the sign:
if(p < 0.5)
x = -x;
T x2 = x * x;
// w is correction term due to skewness
T w = x + sk * (x2 - 1) / 6;
/*
// Add on correction due to kurtosis.
// Disabled for now, seems to make things worse?
//
if(n >= 10)
w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36;
*/
w = m + sigma * w;
if(w < tools::min_value<T>())
return sqrt(tools::min_value<T>());
if(w > n)
return n;
return w;
}

template <class RealType, class Policy>
RealType quantile_imp(const binomial_distribution<RealType, Policy>& dist, const RealType& p, const RealType& q, bool comp)
{ // Quantile or Percent Point Binomial function.
// Return the number of expected successes k,
// for a given probability p.
//
// Error checks:
BOOST_MATH_STD_USING  // ADL of std names
RealType result = 0;
RealType trials = dist.trials();
RealType success_fraction = dist.success_fraction();
if(false == binomial_detail::check_dist_and_prob(
"boost::math::quantile(binomial_distribution<%1%> const&, %1%)",
trials,
success_fraction,
p,
&result, Policy()))
{
return result;
}

// Special cases:
//
if(p == 0)
{  // There may actually be no answer to this question,
// since the probability of zero successes may be non-zero,
// but zero is the best we can do:
return 0;
}
if(p == 1)
{  // Probability of n or fewer successes is always one,
// so n is the most sensible answer here:
return trials;
}
if (p <= pow(1 - success_fraction, trials))
{ // p <= pdf(dist, 0) == cdf(dist, 0)
return 0; // So the only reasonable result is zero.
} // And root finder would fail otherwise.
if(success_fraction == 1)
{  // our formulae break down in this case:
return p > 0.5f ? trials : 0;
}

// Solve for quantile numerically:
//
RealType guess = binomial_detail::inverse_binomial_cornish_fisher(trials, success_fraction, p, q, Policy());
RealType factor = 8;
if(trials > 100)
factor = 1.01f; // guess is pretty accurate
else if((trials > 10) && (trials - 1 > guess) && (guess > 3))
factor = 1.15f; // less accurate but OK.
else if(trials < 10)
{
// pretty inaccurate guess in this area:
if(guess > trials / 64)
{
guess = trials / 4;
factor = 2;
}
else
guess = trials / 1024;
}
else
factor = 2; // trials largish, but in far tails.

typedef typename Policy::discrete_quantile_type discrete_quantile_type;
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
return detail::inverse_discrete_quantile(
dist,
comp ? q : p,
comp,
guess,
factor,
RealType(1),
discrete_quantile_type(),
max_iter);
} // quantile

}

template <class RealType = double, class Policy = policies::policy<> >
class binomial_distribution
{
public:
typedef RealType value_type;
typedef Policy policy_type;

binomial_distribution(RealType n = 1, RealType p = 0.5) : m_n(n), m_p(p)
{ // Default n = 1 is the Bernoulli distribution
// with equal probability of 'heads' or 'tails.
RealType r;
binomial_detail::check_dist(
"boost::math::binomial_distribution<%1%>::binomial_distribution",
m_n,
m_p,
&r, Policy());
} // binomial_distribution constructor.

RealType success_fraction() const
{ // Probability.
return m_p;
}
RealType trials() const
{ // Total number of trials.
return m_n;
}

enum interval_type{
clopper_pearson_exact_interval,
jeffreys_prior_interval
};

//
// Estimation of the success fraction parameter.
// The best estimate is actually simply successes/trials,
// these functions are used
// to obtain confidence intervals for the success fraction.
//
static RealType find_lower_bound_on_p(
RealType trials,
RealType successes,
RealType probability,
interval_type t = clopper_pearson_exact_interval)
{
static const char* function = "boost::math::binomial_distribution<%1%>::find_lower_bound_on_p";
// Error checks:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
function, trials, RealType(0), successes, &result, Policy())
&&
binomial_detail::check_dist_and_prob(
function, trials, RealType(0), probability, &result, Policy()))
{ return result; }

if(successes == 0)
return 0;

// NOTE!!! The Clopper Pearson formula uses "successes" not
// "successes+1" as usual to get the lower bound,
// see http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
return (t == clopper_pearson_exact_interval) ? ibeta_inv(successes, trials - successes + 1, probability, static_cast<RealType*>(0), Policy())
: ibeta_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy());
}
static RealType find_upper_bound_on_p(
RealType trials,
RealType successes,
RealType probability,
interval_type t = clopper_pearson_exact_interval)
{
static const char* function = "boost::math::binomial_distribution<%1%>::find_upper_bound_on_p";
// Error checks:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
function, trials, RealType(0), successes, &result, Policy())
&&
binomial_detail::check_dist_and_prob(
function, trials, RealType(0), probability, &result, Policy()))
{ return result; }

if(trials == successes)
return 1;

return (t == clopper_pearson_exact_interval) ? ibetac_inv(successes + 1, trials - successes, probability, static_cast<RealType*>(0), Policy())
: ibetac_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy());
}
// Estimate number of trials parameter:
//
// "How many trials do I need to be P% sure of seeing k events?"
//    or
// "How many trials can I have to be P% sure of seeing fewer than k events?"
//
static RealType find_minimum_number_of_trials(
RealType k,     // number of events
RealType p,     // success fraction
RealType alpha) // risk level
{
static const char* function = "boost::math::binomial_distribution<%1%>::find_minimum_number_of_trials";
// Error checks:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
function, k, p, k, &result, Policy())
&&
binomial_detail::check_dist_and_prob(
function, k, p, alpha, &result, Policy()))
{ return result; }

result = ibetac_invb(k + 1, p, alpha, Policy());  // returns n - k
return result + k;
}

static RealType find_maximum_number_of_trials(
RealType k,     // number of events
RealType p,     // success fraction
RealType alpha) // risk level
{
static const char* function = "boost::math::binomial_distribution<%1%>::find_maximum_number_of_trials";
// Error checks:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
function, k, p, k, &result, Policy())
&&
binomial_detail::check_dist_and_prob(
function, k, p, alpha, &result, Policy()))
{ return result; }

result = ibeta_invb(k + 1, p, alpha, Policy());  // returns n - k
return result + k;
}

private:
RealType m_n; // Not sure if this shouldn't be an int?
RealType m_p; // success_fraction
}; // template <class RealType, class Policy> class binomial_distribution

typedef binomial_distribution<> binomial;
// typedef binomial_distribution<double> binomial;
// IS now included since no longer a name clash with function binomial.
//typedef binomial_distribution<double> binomial; // Reserved name of type double.

template <class RealType, class Policy>
const std::pair<RealType, RealType> range(const binomial_distribution<RealType, Policy>& dist)
{ // Range of permissible values for random variable k.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(static_cast<RealType>(0), dist.trials());
}

template <class RealType, class Policy>
const std::pair<RealType, RealType> support(const binomial_distribution<RealType, Policy>& dist)
{ // Range of supported values for random variable k.
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
return std::pair<RealType, RealType>(static_cast<RealType>(0),  dist.trials());
}

template <class RealType, class Policy>
inline RealType mean(const binomial_distribution<RealType, Policy>& dist)
{ // Mean of Binomial distribution = np.
return  dist.trials() * dist.success_fraction();
} // mean

template <class RealType, class Policy>
inline RealType variance(const binomial_distribution<RealType, Policy>& dist)
{ // Variance of Binomial distribution = np(1-p).
return  dist.trials() * dist.success_fraction() * (1 - dist.success_fraction());
} // variance

template <class RealType, class Policy>
RealType pdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k)
{ // Probability Density/Mass Function.
BOOST_FPU_EXCEPTION_GUARD

BOOST_MATH_STD_USING // for ADL of std functions

RealType n = dist.trials();

// Error check:
RealType result = 0; // initialization silences some compiler warnings
if(false == binomial_detail::check_dist_and_k(
"boost::math::pdf(binomial_distribution<%1%> const&, %1%)",
n,
dist.success_fraction(),
k,
&result, Policy()))
{
return result;
}

// Special cases of success_fraction, regardless of k successes and regardless of n trials.
if (dist.success_fraction() == 0)
{  // probability of zero successes is 1:
return static_cast<RealType>(k == 0 ? 1 : 0);
}
if (dist.success_fraction() == 1)
{  // probability of n successes is 1:
return static_cast<RealType>(k == n ? 1 : 0);
}
// k argument may be integral, signed, or unsigned, or floating point.
// If necessary, it has already been promoted from an integral type.
if (n == 0)
{
return 1; // Probability = 1 = certainty.
}
if (k == 0)
{ // binomial coeffic (n 0) = 1,
// n ^ 0 = 1
return pow(1 - dist.success_fraction(), n);
}
if (k == n)
{ // binomial coeffic (n n) = 1,
// n ^ 0 = 1
return pow(dist.success_fraction(), k);  // * pow((1 - dist.success_fraction()), (n - k)) = 1
}

// Probability of getting exactly k successes
// if C(n, k) is the binomial coefficient then:
//
// f(k; n,p) = C(n, k) * p^k * (1-p)^(n-k)
//           = (n!/(k!(n-k)!)) * p^k * (1-p)^(n-k)
//           = (tgamma(n+1) / (tgamma(k+1)*tgamma(n-k+1))) * p^k * (1-p)^(n-k)
//           = p^k (1-p)^(n-k) / (beta(k+1, n-k+1) * (n+1))
//           = ibeta_derivative(k+1, n-k+1, p) / (n+1)
//
using boost::math::ibeta_derivative; // a, b, x
return ibeta_derivative(k+1, n-k+1, dist.success_fraction(), Policy()) / (n+1);

} // pdf

template <class RealType, class Policy>
inline RealType cdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k)
{ // Cumulative Distribution Function Binomial.
// The random variate k is the number of successes in n trials.
// k argument may be integral, signed, or unsigned, or floating point.
// If necessary, it has already been promoted from an integral type.

// Returns the sum of the terms 0 through k of the Binomial Probability Density/Mass:
//
//   i=k
//   --  ( n )   i      n-i
//   >   |   |  p  (1-p)
//   --  ( i )
//   i=0

// The terms are not summed directly instead
// the incomplete beta integral is employed,
// according to the formula:
// P = I[1-p]( n-k, k+1).
//   = 1 - I[p](k + 1, n - k)

BOOST_MATH_STD_USING // for ADL of std functions

RealType n = dist.trials();
RealType p = dist.success_fraction();

// Error check:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
"boost::math::cdf(binomial_distribution<%1%> const&, %1%)",
n,
p,
k,
&result, Policy()))
{
return result;
}
if (k == n)
{
return 1;
}

// Special cases, regardless of k.
if (p == 0)
{  // This need explanation:
// the pdf is zero for all cases except when k == 0.
// For zero p the probability of zero successes is one.
// Therefore the cdf is always 1:
// the probability of k or *fewer* successes is always 1
// if there are never any successes!
return 1;
}
if (p == 1)
{ // This is correct but needs explanation:
// when k = 1
// all the cdf and pdf values are zero *except* when k == n,
// and that case has been handled above already.
return 0;
}
//
// P = I[1-p](n - k, k + 1)
//   = 1 - I[p](k + 1, n - k)
// Use of ibetac here prevents cancellation errors in calculating
// 1-p if p is very small, perhaps smaller than machine epsilon.
//
// Note that we do not use a finite sum here, since the incomplete
// beta uses a finite sum internally for integer arguments, so
// we'll just let it take care of the necessary logic.
//
return ibetac(k + 1, n - k, p, Policy());
} // binomial cdf

template <class RealType, class Policy>
inline RealType cdf(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c)
{ // Complemented Cumulative Distribution Function Binomial.
// The random variate k is the number of successes in n trials.
// k argument may be integral, signed, or unsigned, or floating point.
// If necessary, it has already been promoted from an integral type.

// Returns the sum of the terms k+1 through n of the Binomial Probability Density/Mass:
//
//   i=n
//   --  ( n )   i      n-i
//   >   |   |  p  (1-p)
//   --  ( i )
//   i=k+1

// The terms are not summed directly instead
// the incomplete beta integral is employed,
// according to the formula:
// Q = 1 -I[1-p]( n-k, k+1).
//   = I[p](k + 1, n - k)

BOOST_MATH_STD_USING // for ADL of std functions

RealType const& k = c.param;
binomial_distribution<RealType, Policy> const& dist = c.dist;
RealType n = dist.trials();
RealType p = dist.success_fraction();

// Error checks:
RealType result = 0;
if(false == binomial_detail::check_dist_and_k(
"boost::math::cdf(binomial_distribution<%1%> const&, %1%)",
n,
p,
k,
&result, Policy()))
{
return result;
}

if (k == n)
{ // Probability of greater than n successes is necessarily zero:
return 0;
}

// Special cases, regardless of k.
if (p == 0)
{
// This need explanation: the pdf is zero for all
// cases except when k == 0.  For zero p the probability
// of zero successes is one.  Therefore the cdf is always
// 1: the probability of *more than* k successes is always 0
// if there are never any successes!
return 0;
}
if (p == 1)
{
// This needs explanation, when p = 1
// we always have n successes, so the probability
// of more than k successes is 1 as long as k < n.
// The k == n case has already been handled above.
return 1;
}
//
// Calculate cdf binomial using the incomplete beta function.
// Q = 1 -I[1-p](n - k, k + 1)
//   = I[p](k + 1, n - k)
// Use of ibeta here prevents cancellation errors in calculating
// 1-p if p is very small, perhaps smaller than machine epsilon.
//
// Note that we do not use a finite sum here, since the incomplete
// beta uses a finite sum internally for integer arguments, so
// we'll just let it take care of the necessary logic.
//
return ibeta(k + 1, n - k, p, Policy());
} // binomial cdf

template <class RealType, class Policy>
inline RealType quantile(const binomial_distribution<RealType, Policy>& dist, const RealType& p)
{
return binomial_detail::quantile_imp(dist, p, RealType(1-p), false);
} // quantile

template <class RealType, class Policy>
RealType quantile(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c)
{
return binomial_detail::quantile_imp(c.dist, RealType(1-c.param), c.param, true);
} // quantile

template <class RealType, class Policy>
inline RealType mode(const binomial_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING // ADL of std functions.
RealType p = dist.success_fraction();
RealType n = dist.trials();
return floor(p * (n + 1));
}

template <class RealType, class Policy>
inline RealType median(const binomial_distribution<RealType, Policy>& dist)
{ // Bounds for the median of the negative binomial distribution
// VAN DE VEN R. ; WEBER N. C. ;
// Univ. Sydney, school mathematics statistics, Sydney N.S.W. 2006, AUSTRALIE
// Metrika  (Metrika)  ISSN 0026-1335   CODEN MTRKA8
// 1993, vol. 40, no3-4, pp. 185-189 (4 ref.)

// Bounds for median and 50 percentage point of binomial and negative binomial distribution
// Metrika, ISSN   0026-1335 (Print) 1435-926X (Online)
// Volume 41, Number 1 / December, 1994, DOI   10.1007/BF01895303
BOOST_MATH_STD_USING // ADL of std functions.
RealType p = dist.success_fraction();
RealType n = dist.trials();
// Wikipedia says one of floor(np) -1, floor (np), floor(np) +1
return floor(p * n); // Chose the middle value.
}

template <class RealType, class Policy>
inline RealType skewness(const binomial_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING // ADL of std functions.
RealType p = dist.success_fraction();
RealType n = dist.trials();
return (1 - 2 * p) / sqrt(n * p * (1 - p));
}

template <class RealType, class Policy>
inline RealType kurtosis(const binomial_distribution<RealType, Policy>& dist)
{
RealType p = dist.success_fraction();
RealType n = dist.trials();
return 3 - 6 / n + 1 / (n * p * (1 - p));
}

template <class RealType, class Policy>
inline RealType kurtosis_excess(const binomial_distribution<RealType, Policy>& dist)
{
RealType p = dist.success_fraction();
RealType q = 1 - p;
RealType n = dist.trials();
return (1 - 6 * p * q) / (n * p * q);
}

} // namespace math
} // namespace boost

// This include must be at the end, *after* the accessors
// for this distribution have been defined, in order to
// keep compilers that support two-phase lookup happy.
#include <boost/math/distributions/detail/derived_accessors.hpp>

#endif // BOOST_MATH_SPECIAL_BINOMIAL_HPP

```