# Boost C++ Libraries

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### boost/math/distributions/geometric.hpp

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

// Copyright Paul A. Bristow 2010.

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

// 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://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 = 0;  // 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,
// 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 = 0;  // 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,
// 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 = 0;
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 = 0;
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 = 0;
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 = 0;
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, Policy()) * (k + 1);
RealType probability = -boost::math::expm1(z, Policy());

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 = 0;
if(false == geometric_detail::check_dist_and_k(
function,
p,
k,
&result, Policy()))
{
return result;
}
RealType z = boost::math::log1p(-p, Policy()) * (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 = 0;
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, Policy()) / boost::math::log1p(-success_fraction, Policy()) - 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 = 0;
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, Policy()) - 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
```