comparison DEPENDENCIES/generic/include/boost/math/distributions/binomial.hpp @ 16:2665513ce2d3

Add boost headers

+// boost\math\distributions\binomial.hpp
+
+// Copyright John Maddock 2006.
+// Copyright Paul A. Bristow 2007.
+
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0.
+// (See accompanying file LICENSE_1_0.txt
+// or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+// 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.
+// The number of roadkill found per unit length of road
+
+// 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 continous 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 percetage 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
+
+