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 // 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 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/tools/config.hpp>
 #include <boost/math/tools/tuple.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>
         BOOST_MATH_CUDA_ENABLED 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>
         BOOST_MATH_CUDA_ENABLED 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>
         BOOST_MATH_CUDA_ENABLED 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>
         BOOST_MATH_CUDA_ENABLED 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>
         BOOST_MATH_CUDA_ENABLED 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>
          BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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 || success_fraction == 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.
 
         // 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;
         std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
         result = detail::inverse_discrete_quantile(
             dist,
             comp ? q : p,
             comp,
             guess,
             factor,
             RealType(1),
             discrete_quantile_type(),
             max_iter);
         return result;
       } // 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.
 
       BOOST_MATH_CUDA_ENABLED RealType success_fraction() const
       { // Probability.
         return m_p;
       }
       BOOST_MATH_CUDA_ENABLED 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.
       //
       BOOST_MATH_CUDA_ENABLED static RealType find_lower_bound_on_p(
          RealType trials,
          RealType successes,
          RealType probability,
          interval_type t = clopper_pearson_exact_interval)
       {
         BOOST_MATH_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*>(nullptr), Policy())
            : ibeta_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(nullptr), Policy());
       }
       BOOST_MATH_CUDA_ENABLED static RealType find_upper_bound_on_p(
          RealType trials,
          RealType successes,
          RealType probability,
          interval_type t = clopper_pearson_exact_interval)
       {
         BOOST_MATH_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*>(nullptr), Policy())
            : ibetac_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(nullptr), 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?"
       //
       BOOST_MATH_CUDA_ENABLED static RealType find_minimum_number_of_trials(
          RealType k,     // number of events
          RealType p,     // success fraction
          RealType alpha) // risk level
       {
         BOOST_MATH_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;
       }
 
       BOOST_MATH_CUDA_ENABLED static RealType find_maximum_number_of_trials(
          RealType k,     // number of events
          RealType p,     // success fraction
          RealType alpha) // risk level
       {
         BOOST_MATH_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.
 
       #ifdef __cpp_deduction_guides
       template <class RealType>
       binomial_distribution(RealType)->binomial_distribution<typename boost::math::tools::promote_args<RealType>::type>;
       template <class RealType>
       binomial_distribution(RealType,RealType)->binomial_distribution<typename boost::math::tools::promote_args<RealType>::type>;
       #endif
 
       template <class RealType, class Policy>
       BOOST_MATH_CUDA_ENABLED const boost::math::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 boost::math::pair<RealType, RealType>(static_cast<RealType>(0), dist.trials());
       }
 
       template <class RealType, class Policy>
       BOOST_MATH_CUDA_ENABLED const boost::math::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 boost::math::pair<RealType, RealType>(static_cast<RealType>(0),  dist.trials());
       }
 
       template <class RealType, class Policy>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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 == 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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>
       BOOST_MATH_CUDA_ENABLED 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
 
 

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