third_party/boost/include/boost/math/distributions/geometric.hpp - webm/webmlive - Git at Google

 // boost\math\distributions\geometric.hpp

 // Copyright John Maddock 2010.
 // Copyright Paul A. Bristow 2010.

 // 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)

 // geometric distribution is a discrete probability distribution.
 // It expresses the probability distribution of the number (k) of
 // events, occurrences, failures or arrivals before the first success.
 // supported on the set {0, 1, 2, 3...}

 // Note that the set includes zero (unlike some definitions that start at one).

 // The random variate k is the number of events, occurrences or arrivals.
 // k argument may be integral, signed, or unsigned, or floating point.
 // If necessary, it has already been promoted from an integral type.

 // Note that the geometric distribution
 // (like others including the binomial, geometric & Bernoulli)
 // is strictly defined as a discrete function:
 // only integral values of k are envisaged.
 // However because the method of calculation uses 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.

 // See http://en.wikipedia.org/wiki/geometric_distribution
 // http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html
 // http://mathworld.wolfram.com/GeometricDistribution.html

 #ifndef BOOST_MATH_SPECIAL_GEOMETRIC_HPP
 #define BOOST_MATH_SPECIAL_GEOMETRIC_HPP

 #include <boost/math/distributions/fwd.hpp>
 #include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x) == Ix(a, b).
 #include <boost/math/distributions/complement.hpp> // complement.
 #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks domain_error & logic_error.
 #include <boost/math/special_functions/fpclassify.hpp> // isnan.
 #include <boost/math/tools/roots.hpp> // for root finding.
 #include <boost/math/distributions/detail/inv_discrete_quantile.hpp>

 #include <boost/type_traits/is_floating_point.hpp>
 #include <boost/type_traits/is_integral.hpp>
 #include <boost/type_traits/is_same.hpp>
 #include <boost/mpl/if.hpp>

 #include <limits> // using std::numeric_limits;
 #include <utility>

 #if defined (BOOST_MSVC)
 #  pragma warning(push)
 // This believed not now necessary, so commented out.
 //#  pragma warning(disable: 4702) // unreachable code.
 // in domain_error_imp in error_handling.
 #endif

 namespace boost
 {
   namespace math
   {
     namespace geometric_detail
     {
       // Common error checking routines for geometric distribution function:
       template <class RealType, class Policy>
       inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)
       {
         if( !(boost::math::isfinite)(p) || (p < 0) || (p > 1) )
         {
           *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& p, RealType* result, const Policy& pol)
       {
         return check_success_fraction(function, p, result, pol);
       }

       template <class RealType, class Policy>
       inline bool check_dist_and_k(const char* function,  const RealType& p, RealType k, RealType* result, const Policy& pol)
       {
         if(check_dist(function, p, result, pol) == false)
         {
           return false;
         }
         if( !(boost::math::isfinite)(k) || (k < 0) )
         { // Check k failures.
           *result = policies::raise_domain_error<RealType>(
             function,
             "Number of failures argument is %1%, but must be >= 0 !", k, pol);
           return false;
         }
         return true;
       } // Check_dist_and_k

       template <class RealType, class Policy>
       inline bool check_dist_and_prob(const char* function, RealType p, RealType prob, RealType* result, const Policy& pol)
       {
         if(check_dist(function, p, result, pol) && detail::check_probability(function, prob, result, pol) == false)
         {
           return false;
         }
         return true;
       } // check_dist_and_prob
     } //  namespace geometric_detail

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

       geometric_distribution(RealType p) : m_p(p)
       { // Constructor stores success_fraction p.
         RealType result;
         geometric_detail::check_dist(
           "geometric_distribution<%1%>::geometric_distribution",
           m_p, // Check success_fraction 0 <= p <= 1.
           &result, Policy());
       } // geometric_distribution constructor.

       // Private data getter class member functions.
       RealType success_fraction() const
       { // Probability of success as fraction in range 0 to 1.
         return m_p;
       }
       RealType successes() const
       { // Total number of successes r = 1 (for compatibility with negative binomial?).
         return 1;
       }

       // Parameter estimation.
       // (These are copies of negative_binomial distribution with successes = 1).
       static RealType find_lower_bound_on_p(
         RealType trials,
         RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
       {
         static const char* function = "boost::math::geometric<%1%>::find_lower_bound_on_p";
         RealType result;  // of error checks.
         RealType successes = 1;
         RealType failures = trials - successes;
         if(false == detail::check_probability(function, alpha, &result, Policy())
           && geometric_detail::check_dist_and_k(
           function, RealType(0), failures, &result, Policy()))
         {
           return result;
         }
         // Use complement ibeta_inv function for lower bound.
         // This is adapted from the corresponding binomial formula
         // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
         // This is a Clopper-Pearson interval, and may be overly conservative,
         // see also "A Simple Improved Inferential Method for Some
         // Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY
         // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
         //
         return ibeta_inv(successes, failures + 1, alpha, static_cast<RealType*>(0), Policy());
       } // find_lower_bound_on_p

       static RealType find_upper_bound_on_p(
         RealType trials,
         RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
       {
         static const char* function = "boost::math::geometric<%1%>::find_upper_bound_on_p";
         RealType result;  // of error checks.
         RealType successes = 1;
         RealType failures = trials - successes;
         if(false == geometric_detail::check_dist_and_k(
           function, RealType(0), failures, &result, Policy())
           && detail::check_probability(function, alpha, &result, Policy()))
         {
           return result;
         }
         if(failures == 0)
         {
            return 1;
         }// Use complement ibetac_inv function for upper bound.
         // Note adjusted failures value: *not* failures+1 as usual.
         // This is adapted from the corresponding binomial formula
         // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
         // This is a Clopper-Pearson interval, and may be overly conservative,
         // see also "A Simple Improved Inferential Method for Some
         // Discrete Distributions" Yong CAI and K. Krishnamoorthy
         // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
         //
         return ibetac_inv(successes, failures, alpha, static_cast<RealType*>(0), Policy());
       } // find_upper_bound_on_p

       // Estimate number of trials :
       // "How many trials do I need to be P% sure of seeing k or fewer failures?"

       static RealType find_minimum_number_of_trials(
         RealType k,     // number of failures (k >= 0).
         RealType p,     // success fraction 0 <= p <= 1.
         RealType alpha) // risk level threshold 0 <= alpha <= 1.
       {
         static const char* function = "boost::math::geometric<%1%>::find_minimum_number_of_trials";
         // Error checks:
         RealType result;
         if(false == geometric_detail::check_dist_and_k(
           function, p, k, &result, Policy())
           && detail::check_probability(function, alpha, &result, Policy()))
         {
           return result;
         }
         result = ibeta_inva(k + 1, p, alpha, Policy());  // returns n - k
         return result + k;
       } // RealType find_number_of_failures

       static RealType find_maximum_number_of_trials(
         RealType k,     // number of failures (k >= 0).
         RealType p,     // success fraction 0 <= p <= 1.
         RealType alpha) // risk level threshold 0 <= alpha <= 1.
       {
         static const char* function = "boost::math::geometric<%1%>::find_maximum_number_of_trials";
         // Error checks:
         RealType result;
         if(false == geometric_detail::check_dist_and_k(
           function, p, k, &result, Policy())
           &&  detail::check_probability(function, alpha, &result, Policy()))
         {
           return result;
         }
         result = ibetac_inva(k + 1, p, alpha, Policy());  // returns n - k
         return result + k;
       } // RealType find_number_of_trials complemented

     private:
       //RealType m_r; // successes fixed at unity.
       RealType m_p; // success_fraction
     }; // template <class RealType, class Policy> class geometric_distribution

     typedef geometric_distribution<double> geometric; // Reserved name of type double.

     template <class RealType, class Policy>
     inline const std::pair<RealType, RealType> range(const geometric_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), max_value<RealType>()); // max_integer?
     }

     template <class RealType, class Policy>
     inline const std::pair<RealType, RealType> support(const geometric_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.
        using boost::math::tools::max_value;
        return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>()); // max_integer?
     }

     template <class RealType, class Policy>
     inline RealType mean(const geometric_distribution<RealType, Policy>& dist)
     { // Mean of geometric distribution = (1-p)/p.
       return (1 - dist.success_fraction() ) / dist.success_fraction();
     } // mean

     // median implemented via quantile(half) in derived accessors.

     template <class RealType, class Policy>
     inline RealType mode(const geometric_distribution<RealType, Policy>&)
     { // Mode of geometric distribution = zero.
       BOOST_MATH_STD_USING // ADL of std functions.
       return 0;
     } // mode

     template <class RealType, class Policy>
     inline RealType variance(const geometric_distribution<RealType, Policy>& dist)
     { // Variance of Binomial distribution = (1-p) / p^2.
       return  (1 - dist.success_fraction())
         / (dist.success_fraction() * dist.success_fraction());
     } // variance

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

     template <class RealType, class Policy>
     inline RealType kurtosis(const geometric_distribution<RealType, Policy>& dist)
     { // kurtosis of geometric distribution
       // http://en.wikipedia.org/wiki/geometric is kurtosis_excess so add 3
       RealType p = dist.success_fraction();
       return 3 + (p*p - 6*p + 6) / (1 - p);
     } // kurtosis

      template <class RealType, class Policy>
     inline RealType kurtosis_excess(const geometric_distribution<RealType, Policy>& dist)
     { // kurtosis excess of geometric distribution
       // http://mathworld.wolfram.com/Kurtosis.html table of kurtosis_excess
       RealType p = dist.success_fraction();
       return (p*p - 6*p + 6) / (1 - p);
     } // kurtosis_excess

     // RealType standard_deviation(const geometric_distribution<RealType, Policy>& dist)
     // standard_deviation provided by derived accessors.
     // RealType hazard(const geometric_distribution<RealType, Policy>& dist)
     // hazard of geometric distribution provided by derived accessors.
     // RealType chf(const geometric_distribution<RealType, Policy>& dist)
     // chf of geometric distribution provided by derived accessors.

     template <class RealType, class Policy>
     inline RealType pdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
     { // Probability Density/Mass Function.
       BOOST_FPU_EXCEPTION_GUARD
       BOOST_MATH_STD_USING  // For ADL of math functions.
       static const char* function = "boost::math::pdf(const geometric_distribution<%1%>&, %1%)";

       RealType p = dist.success_fraction();
       RealType result;
       if(false == geometric_detail::check_dist_and_k(
         function,
         p,
         k,
         &result, Policy()))
       {
         return result;
       }
       if (k == 0)
       {
         return p; // success_fraction
       }
       RealType q = 1 - p;  // Inaccurate for small p?
       // So try to avoid inaccuracy for large or small p.
       // but has little effect > last significant bit.
       //cout << "p *  pow(q, k) " << result << endl; // seems best whatever p
       //cout << "exp(p * k * log1p(-p)) " << p * exp(k * log1p(-p)) << endl;
       //if (p < 0.5)
       //{
       //  result = p *  pow(q, k);
       //}
       //else
       //{
       //  result = p * exp(k * log1p(-p));
       //}
       result = p * pow(q, k);
       return result;
     } // geometric_pdf

     template <class RealType, class Policy>
     inline RealType cdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
     { // Cumulative Distribution Function of geometric.
       static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";

       // k argument may be integral, signed, or unsigned, or floating point.
       // If necessary, it has already been promoted from an integral type.
       RealType p = dist.success_fraction();
       // Error check:
       RealType result;
       if(false == geometric_detail::check_dist_and_k(
         function,
         p,
         k,
         &result, Policy()))
       {
         return result;
       }
       if(k == 0)
       {
         return p; // success_fraction
       }
       //RealType q = 1 - p;  // Bad for small p
       //RealType probability = 1 - std::pow(q, k+1);

       RealType z = boost::math::log1p(-p) * (k+1);
       RealType probability = -boost::math::expm1(z);

       return probability;
     } // cdf Cumulative Distribution Function geometric.

       template <class RealType, class Policy>
       inline RealType cdf(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
       { // Complemented Cumulative Distribution Function geometric.
       BOOST_MATH_STD_USING
       static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";
       // k argument may be integral, signed, or unsigned, or floating point.
       // If necessary, it has already been promoted from an integral type.
       RealType const& k = c.param;
       geometric_distribution<RealType, Policy> const& dist = c.dist;
       RealType p = dist.success_fraction();
       // Error check:
       RealType result;
       if(false == geometric_detail::check_dist_and_k(
         function,
         p,
         k,
         &result, Policy()))
       {
         return result;
       }
       RealType z = boost::math::log1p(-p) * (k+1);
       RealType probability = exp(z);
       return probability;
     } // cdf Complemented Cumulative Distribution Function geometric.

     template <class RealType, class Policy>
     inline RealType quantile(const geometric_distribution<RealType, Policy>& dist, const RealType& x)
     { // Quantile, percentile/100 or Percent Point geometric function.
       // Return the number of expected failures k for a given probability p.

       // Inverse cumulative Distribution Function or Quantile (percentile / 100) of geometric Probability.
       // k argument may be integral, signed, or unsigned, or floating point.

       static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
       BOOST_MATH_STD_USING // ADL of std functions.

       RealType success_fraction = dist.success_fraction();
       // Check dist and x.
       RealType result;
       if(false == geometric_detail::check_dist_and_prob
         (function, success_fraction, x, &result, Policy()))
       {
         return result;
       }

       // Special cases.
       if (x == 1)
       {  // Would need +infinity failures for total confidence.
         result = policies::raise_overflow_error<RealType>(
             function,
             "Probability argument is 1, which implies infinite failures !", Policy());
         return result;
        // usually means return +std::numeric_limits<RealType>::infinity();
        // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
       }
       if (x == 0)
       { // No failures are expected if P = 0.
         return 0; // Total trials will be just dist.successes.
       }
       // if (P <= pow(dist.success_fraction(), 1))
       if (x <= success_fraction)
       { // p <= pdf(dist, 0) == cdf(dist, 0)
         return 0;
       }
       if (x == 1)
       {
         return 0;
       }

       // log(1-x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
       result = boost::math::log1p(-x) / boost::math::log1p(-success_fraction) -1;
       // Subtract a few epsilons here too?
       // to make sure it doesn't slip over, so ceil would be one too many.
       return result;
     } // RealType quantile(const geometric_distribution dist, p)

     template <class RealType, class Policy>
     inline RealType quantile(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
     {  // Quantile or Percent Point Binomial function.
        // Return the number of expected failures k for a given
        // complement of the probability Q = 1 - P.
        static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
        BOOST_MATH_STD_USING
        // Error checks:
        RealType x = c.param;
        const geometric_distribution<RealType, Policy>& dist = c.dist;
        RealType success_fraction = dist.success_fraction();
        RealType result;
        if(false == geometric_detail::check_dist_and_prob(
           function,
           success_fraction,
           x,
           &result, Policy()))
        {
           return result;
        }

        // Special cases:
        if(x == 1)
        {  // There may actually be no answer to this question,
           // since the probability of zero failures may be non-zero,
           return 0; // but zero is the best we can do:
        }
        if (-x <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))
        {  // q <= cdf(complement(dist, 0)) == pdf(dist, 0)
           return 0; //
        }
        if(x == 0)
        {  // Probability 1 - Q  == 1 so infinite failures to achieve certainty.
           // Would need +infinity failures for total confidence.
           result = policies::raise_overflow_error<RealType>(
              function,
              "Probability argument complement is 0, which implies infinite failures !", Policy());
           return result;
           // usually means return +std::numeric_limits<RealType>::infinity();
           // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
        }
        // log(x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
        result = log(x) / boost::math::log1p(-success_fraction) -1;
       return result;

     } // quantile complement

  } // 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>

 #if defined (BOOST_MSVC)
 # pragma warning(pop)
 #endif

 #endif // BOOST_MATH_SPECIAL_GEOMETRIC_HPP
	// boost\math\distributions\geometric.hpp

	// Copyright John Maddock 2010.
	// Copyright Paul A. Bristow 2010.

	// 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)

	// geometric distribution is a discrete probability distribution.
	// It expresses the probability distribution of the number (k) of
	// events, occurrences, failures or arrivals before the first success.
	// supported on the set {0, 1, 2, 3...}

	// Note that the set includes zero (unlike some definitions that start at one).

	// The random variate k is the number of events, occurrences or arrivals.
	// k argument may be integral, signed, or unsigned, or floating point.
	// If necessary, it has already been promoted from an integral type.

	// Note that the geometric distribution
	// (like others including the binomial, geometric & Bernoulli)
	// is strictly defined as a discrete function:
	// only integral values of k are envisaged.
	// However because the method of calculation uses 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.

	// See http://en.wikipedia.org/wiki/geometric_distribution
	// http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html
	// http://mathworld.wolfram.com/GeometricDistribution.html

	#ifndef BOOST_MATH_SPECIAL_GEOMETRIC_HPP
	#define BOOST_MATH_SPECIAL_GEOMETRIC_HPP

	#include <boost/math/distributions/fwd.hpp>
	#include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x) == Ix(a, b).
	#include <boost/math/distributions/complement.hpp> // complement.
	#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks domain_error & logic_error.
	#include <boost/math/special_functions/fpclassify.hpp> // isnan.
	#include <boost/math/tools/roots.hpp> // for root finding.
	#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>

	#include <boost/type_traits/is_floating_point.hpp>
	#include <boost/type_traits/is_integral.hpp>
	#include <boost/type_traits/is_same.hpp>
	#include <boost/mpl/if.hpp>

	#include <limits> // using std::numeric_limits;
	#include <utility>

	#if defined (BOOST_MSVC)
	# pragma warning(push)
	// This believed not now necessary, so commented out.
	//# pragma warning(disable: 4702) // unreachable code.
	// in domain_error_imp in error_handling.
	#endif

	namespace boost
	{
	namespace math
	{
	namespace geometric_detail
	{
	// Common error checking routines for geometric distribution function:
	template <class RealType, class Policy>
	inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)
	{
	if( !(boost::math::isfinite)(p) \|\| (p < 0) \|\| (p > 1) )
	{
	*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& p, RealType* result, const Policy& pol)
	{
	return check_success_fraction(function, p, result, pol);
	}

	template <class RealType, class Policy>
	inline bool check_dist_and_k(const char* function, const RealType& p, RealType k, RealType* result, const Policy& pol)
	{
	if(check_dist(function, p, result, pol) == false)
	{
	return false;
	}
	if( !(boost::math::isfinite)(k) \|\| (k < 0) )
	{ // Check k failures.
	*result = policies::raise_domain_error<RealType>(
	function,
	"Number of failures argument is %1%, but must be >= 0 !", k, pol);
	return false;
	}
	return true;
	} // Check_dist_and_k

	template <class RealType, class Policy>
	inline bool check_dist_and_prob(const char* function, RealType p, RealType prob, RealType* result, const Policy& pol)
	{
	if(check_dist(function, p, result, pol) && detail::check_probability(function, prob, result, pol) == false)
	{
	return false;
	}
	return true;
	} // check_dist_and_prob
	} // namespace geometric_detail

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

	geometric_distribution(RealType p) : m_p(p)
	{ // Constructor stores success_fraction p.
	RealType result;
	geometric_detail::check_dist(
	"geometric_distribution<%1%>::geometric_distribution",
	m_p, // Check success_fraction 0 <= p <= 1.
	&result, Policy());
	} // geometric_distribution constructor.

	// Private data getter class member functions.
	RealType success_fraction() const
	{ // Probability of success as fraction in range 0 to 1.
	return m_p;
	}
	RealType successes() const
	{ // Total number of successes r = 1 (for compatibility with negative binomial?).
	return 1;
	}

	// Parameter estimation.
	// (These are copies of negative_binomial distribution with successes = 1).
	static RealType find_lower_bound_on_p(
	RealType trials,
	RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
	{
	static const char* function = "boost::math::geometric<%1%>::find_lower_bound_on_p";
	RealType result; // of error checks.
	RealType successes = 1;
	RealType failures = trials - successes;
	if(false == detail::check_probability(function, alpha, &result, Policy())
	&& geometric_detail::check_dist_and_k(
	function, RealType(0), failures, &result, Policy()))
	{
	return result;
	}
	// Use complement ibeta_inv function for lower bound.
	// This is adapted from the corresponding binomial formula
	// here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
	// This is a Clopper-Pearson interval, and may be overly conservative,
	// see also "A Simple Improved Inferential Method for Some
	// Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY
	// http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
	//
	return ibeta_inv(successes, failures + 1, alpha, static_cast<RealType*>(0), Policy());
	} // find_lower_bound_on_p

	static RealType find_upper_bound_on_p(
	RealType trials,
	RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
	{
	static const char* function = "boost::math::geometric<%1%>::find_upper_bound_on_p";
	RealType result; // of error checks.
	RealType successes = 1;
	RealType failures = trials - successes;
	if(false == geometric_detail::check_dist_and_k(
	function, RealType(0), failures, &result, Policy())
	&& detail::check_probability(function, alpha, &result, Policy()))
	{
	return result;
	}
	if(failures == 0)
	{
	return 1;
	}// Use complement ibetac_inv function for upper bound.
	// Note adjusted failures value: not failures+1 as usual.
	// This is adapted from the corresponding binomial formula
	// here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
	// This is a Clopper-Pearson interval, and may be overly conservative,
	// see also "A Simple Improved Inferential Method for Some
	// Discrete Distributions" Yong CAI and K. Krishnamoorthy
	// http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
	//
	return ibetac_inv(successes, failures, alpha, static_cast<RealType*>(0), Policy());
	} // find_upper_bound_on_p

	// Estimate number of trials :
	// "How many trials do I need to be P% sure of seeing k or fewer failures?"

	static RealType find_minimum_number_of_trials(
	RealType k, // number of failures (k >= 0).
	RealType p, // success fraction 0 <= p <= 1.
	RealType alpha) // risk level threshold 0 <= alpha <= 1.
	{
	static const char* function = "boost::math::geometric<%1%>::find_minimum_number_of_trials";
	// Error checks:
	RealType result;
	if(false == geometric_detail::check_dist_and_k(
	function, p, k, &result, Policy())
	&& detail::check_probability(function, alpha, &result, Policy()))
	{
	return result;
	}
	result = ibeta_inva(k + 1, p, alpha, Policy()); // returns n - k
	return result + k;
	} // RealType find_number_of_failures

	static RealType find_maximum_number_of_trials(
	RealType k, // number of failures (k >= 0).
	RealType p, // success fraction 0 <= p <= 1.
	RealType alpha) // risk level threshold 0 <= alpha <= 1.
	{
	static const char* function = "boost::math::geometric<%1%>::find_maximum_number_of_trials";
	// Error checks:
	RealType result;
	if(false == geometric_detail::check_dist_and_k(
	function, p, k, &result, Policy())
	&& detail::check_probability(function, alpha, &result, Policy()))
	{
	return result;
	}
	result = ibetac_inva(k + 1, p, alpha, Policy()); // returns n - k
	return result + k;
	} // RealType find_number_of_trials complemented

	private:
	//RealType m_r; // successes fixed at unity.
	RealType m_p; // success_fraction
	}; // template <class RealType, class Policy> class geometric_distribution

	typedef geometric_distribution<double> geometric; // Reserved name of type double.

	template <class RealType, class Policy>
	inline const std::pair<RealType, RealType> range(const geometric_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), max_value<RealType>()); // max_integer?
	}

	template <class RealType, class Policy>
	inline const std::pair<RealType, RealType> support(const geometric_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.
	using boost::math::tools::max_value;
	return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?
	}

	template <class RealType, class Policy>
	inline RealType mean(const geometric_distribution<RealType, Policy>& dist)
	{ // Mean of geometric distribution = (1-p)/p.
	return (1 - dist.success_fraction() ) / dist.success_fraction();
	} // mean

	// median implemented via quantile(half) in derived accessors.

	template <class RealType, class Policy>
	inline RealType mode(const geometric_distribution<RealType, Policy>&)
	{ // Mode of geometric distribution = zero.
	BOOST_MATH_STD_USING // ADL of std functions.
	return 0;
	} // mode

	template <class RealType, class Policy>
	inline RealType variance(const geometric_distribution<RealType, Policy>& dist)
	{ // Variance of Binomial distribution = (1-p) / p^2.
	return (1 - dist.success_fraction())
	/ (dist.success_fraction() * dist.success_fraction());
	} // variance

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

	template <class RealType, class Policy>
	inline RealType kurtosis(const geometric_distribution<RealType, Policy>& dist)
	{ // kurtosis of geometric distribution
	// http://en.wikipedia.org/wiki/geometric is kurtosis_excess so add 3
	RealType p = dist.success_fraction();
	return 3 + (pp - 6p + 6) / (1 - p);
	} // kurtosis

	template <class RealType, class Policy>
	inline RealType kurtosis_excess(const geometric_distribution<RealType, Policy>& dist)
	{ // kurtosis excess of geometric distribution
	// http://mathworld.wolfram.com/Kurtosis.html table of kurtosis_excess
	RealType p = dist.success_fraction();
	return (pp - 6p + 6) / (1 - p);
	} // kurtosis_excess

	// RealType standard_deviation(const geometric_distribution<RealType, Policy>& dist)
	// standard_deviation provided by derived accessors.
	// RealType hazard(const geometric_distribution<RealType, Policy>& dist)
	// hazard of geometric distribution provided by derived accessors.
	// RealType chf(const geometric_distribution<RealType, Policy>& dist)
	// chf of geometric distribution provided by derived accessors.

	template <class RealType, class Policy>
	inline RealType pdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
	{ // Probability Density/Mass Function.
	BOOST_FPU_EXCEPTION_GUARD
	BOOST_MATH_STD_USING // For ADL of math functions.
	static const char* function = "boost::math::pdf(const geometric_distribution<%1%>&, %1%)";

	RealType p = dist.success_fraction();
	RealType result;
	if(false == geometric_detail::check_dist_and_k(
	function,
	p,
	k,
	&result, Policy()))
	{
	return result;
	}
	if (k == 0)
	{
	return p; // success_fraction
	}
	RealType q = 1 - p; // Inaccurate for small p?
	// So try to avoid inaccuracy for large or small p.
	// but has little effect > last significant bit.
	//cout << "p * pow(q, k) " << result << endl; // seems best whatever p
	//cout << "exp(p * k * log1p(-p)) " << p * exp(k * log1p(-p)) << endl;
	//if (p < 0.5)
	//{
	// result = p * pow(q, k);
	//}
	//else
	//{
	// result = p * exp(k * log1p(-p));
	//}
	result = p * pow(q, k);
	return result;
	} // geometric_pdf

	template <class RealType, class Policy>
	inline RealType cdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
	{ // Cumulative Distribution Function of geometric.
	static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";

	// k argument may be integral, signed, or unsigned, or floating point.
	// If necessary, it has already been promoted from an integral type.
	RealType p = dist.success_fraction();
	// Error check:
	RealType result;
	if(false == geometric_detail::check_dist_and_k(
	function,
	p,
	k,
	&result, Policy()))
	{
	return result;
	}
	if(k == 0)
	{
	return p; // success_fraction
	}
	//RealType q = 1 - p; // Bad for small p
	//RealType probability = 1 - std::pow(q, k+1);

	RealType z = boost::math::log1p(-p) * (k+1);
	RealType probability = -boost::math::expm1(z);

	return probability;
	} // cdf Cumulative Distribution Function geometric.

	template <class RealType, class Policy>
	inline RealType cdf(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
	{ // Complemented Cumulative Distribution Function geometric.
	BOOST_MATH_STD_USING
	static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";
	// k argument may be integral, signed, or unsigned, or floating point.
	// If necessary, it has already been promoted from an integral type.
	RealType const& k = c.param;
	geometric_distribution<RealType, Policy> const& dist = c.dist;
	RealType p = dist.success_fraction();
	// Error check:
	RealType result;
	if(false == geometric_detail::check_dist_and_k(
	function,
	p,
	k,
	&result, Policy()))
	{
	return result;
	}
	RealType z = boost::math::log1p(-p) * (k+1);
	RealType probability = exp(z);
	return probability;
	} // cdf Complemented Cumulative Distribution Function geometric.

	template <class RealType, class Policy>
	inline RealType quantile(const geometric_distribution<RealType, Policy>& dist, const RealType& x)
	{ // Quantile, percentile/100 or Percent Point geometric function.
	// Return the number of expected failures k for a given probability p.

	// Inverse cumulative Distribution Function or Quantile (percentile / 100) of geometric Probability.
	// k argument may be integral, signed, or unsigned, or floating point.

	static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
	BOOST_MATH_STD_USING // ADL of std functions.

	RealType success_fraction = dist.success_fraction();
	// Check dist and x.
	RealType result;
	if(false == geometric_detail::check_dist_and_prob
	(function, success_fraction, x, &result, Policy()))
	{
	return result;
	}

	// Special cases.
	if (x == 1)
	{ // Would need +infinity failures for total confidence.
	result = policies::raise_overflow_error<RealType>(
	function,
	"Probability argument is 1, which implies infinite failures !", Policy());
	return result;
	// usually means return +std::numeric_limits<RealType>::infinity();
	// unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
	}
	if (x == 0)
	{ // No failures are expected if P = 0.
	return 0; // Total trials will be just dist.successes.
	}
	// if (P <= pow(dist.success_fraction(), 1))
	if (x <= success_fraction)
	{ // p <= pdf(dist, 0) == cdf(dist, 0)
	return 0;
	}
	if (x == 1)
	{
	return 0;
	}

	// log(1-x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
	result = boost::math::log1p(-x) / boost::math::log1p(-success_fraction) -1;
	// Subtract a few epsilons here too?
	// to make sure it doesn't slip over, so ceil would be one too many.
	return result;
	} // RealType quantile(const geometric_distribution dist, p)

	template <class RealType, class Policy>
	inline RealType quantile(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
	{ // Quantile or Percent Point Binomial function.
	// Return the number of expected failures k for a given
	// complement of the probability Q = 1 - P.
	static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
	BOOST_MATH_STD_USING
	// Error checks:
	RealType x = c.param;
	const geometric_distribution<RealType, Policy>& dist = c.dist;
	RealType success_fraction = dist.success_fraction();
	RealType result;
	if(false == geometric_detail::check_dist_and_prob(
	function,
	success_fraction,
	x,
	&result, Policy()))
	{
	return result;
	}

	// Special cases:
	if(x == 1)
	{ // There may actually be no answer to this question,
	// since the probability of zero failures may be non-zero,
	return 0; // but zero is the best we can do:
	}
	if (-x <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))
	{ // q <= cdf(complement(dist, 0)) == pdf(dist, 0)
	return 0; //
	}
	if(x == 0)
	{ // Probability 1 - Q == 1 so infinite failures to achieve certainty.
	// Would need +infinity failures for total confidence.
	result = policies::raise_overflow_error<RealType>(
	function,
	"Probability argument complement is 0, which implies infinite failures !", Policy());
	return result;
	// usually means return +std::numeric_limits<RealType>::infinity();
	// unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
	}
	// log(x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
	result = log(x) / boost::math::log1p(-success_fraction) -1;
	return result;

	} // quantile complement

	} // 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>

	#if defined (BOOST_MSVC)
	# pragma warning(pop)
	#endif

	#endif // BOOST_MATH_SPECIAL_GEOMETRIC_HPP