beta.hpp

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      RealType m_alpha; // Two parameters of the beta distribution.
      RealType m_beta;
    }; // template <class RealType, class Policy> class beta_distribution

    template <class RealType, class Policy>
    inline const std::pair<RealType, RealType> range(const beta_distribution<RealType, Policy>& /* dist */)
    { // Range of permissible values for random variable x.
      using boost::math::tools::max_value;
      return std::pair<RealType, RealType>(0, 1);
    }

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

    template <class RealType, class Policy>
    inline RealType mean(const beta_distribution<RealType, Policy>& dist)
    { // Mean of beta distribution = np.
      return  dist.alpha() / (dist.alpha() + dist.beta());
    } // mean

    template <class RealType, class Policy>
    inline RealType variance(const beta_distribution<RealType, Policy>& dist)
    { // Variance of beta distribution = np(1-p).
      RealType a = dist.alpha();
      RealType b = dist.beta();
      return  (a * b) / ((a + b ) * (a + b) * (a + b + 1));
    } // variance

    template <class RealType, class Policy>
    inline RealType mode(const beta_distribution<RealType, Policy>& dist)
    {
      static const char* function = "boost::math::mode(beta_distribution<%1%> const&)";

      RealType result;
      if ((dist.alpha() <= 1))
      {
        result = policies::raise_domain_error<RealType>(
          function,
          "mode undefined for alpha = %1%, must be > 1!", dist.alpha(), Policy());
        return result;
      }

      if ((dist.beta() <= 1))
      {
        result = policies::raise_domain_error<RealType>(
          function,
          "mode undefined for beta = %1%, must be > 1!", dist.beta(), Policy());
        return result;
      }
      RealType a = dist.alpha();
      RealType b = dist.beta();
      return (a-1) / (a + b - 2);
    } // mode

    //template <class RealType, class Policy>
    //inline RealType median(const beta_distribution<RealType, Policy>& dist)
    //{ // Median of beta distribution is not defined.
    //  return tools::domain_error<RealType>(function, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());
    //} // median

    //But WILL be provided by the derived accessor as quantile(0.5).

    template <class RealType, class Policy>
    inline RealType skewness(const beta_distribution<RealType, Policy>& dist)
    {
      BOOST_MATH_STD_USING // ADL of std functions.
      RealType a = dist.alpha();
      RealType b = dist.beta();
      return (2 * (b-a) * sqrt(a + b + 1)) / ((a + b + 2) * sqrt(a * b));
    } // skewness

    template <class RealType, class Policy>
    inline RealType kurtosis_excess(const beta_distribution<RealType, Policy>& dist)
    {
      RealType a = dist.alpha();
      RealType b = dist.beta();
      RealType a_2 = a * a;
      RealType n = 6 * (a_2 * a - a_2 * (2 * b - 1) + b * b * (b + 1) - 2 * a * b * (b + 2));
      RealType d = a * b * (a + b + 2) * (a + b + 3);
      return  n / d;
    } // kurtosis_excess

    template <class RealType, class Policy>
    inline RealType kurtosis(const beta_distribution<RealType, Policy>& dist)
    {
      return 3 + kurtosis_excess(dist);
    } // kurtosis

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

      static const char* function = "boost::math::pdf(beta_distribution<%1%> const&, %1%)";

      BOOST_MATH_STD_USING // for ADL of std functions

      RealType a = dist.alpha();
      RealType b = dist.beta();

      // Argument checks:
      RealType result;
      if(false == beta_detail::check_dist_and_x(
        function,
        a, b, x,
        &result, Policy()))
      {
        return result;
      }
      using boost::math::beta;
      return ibeta_derivative(a, b, x, Policy());
    } // pdf

    template <class RealType, class Policy>
    inline RealType cdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
    { // Cumulative Distribution Function beta.
      BOOST_MATH_STD_USING // for ADL of std functions

      static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";

      RealType a = dist.alpha();
      RealType b = dist.beta();

      // Argument checks:
      RealType result;
      if(false == beta_detail::check_dist_and_x(
        function,
        a, b, x,
        &result, Policy()))
      {
        return result;
      }
      // Special cases:
      if (x == 0)
      {
        return 0;
      }
      else if (x == 1)
      {
        return 1;
      }
      return ibeta(a, b, x, Policy());
    } // beta cdf

    template <class RealType, class Policy>
    inline RealType cdf(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
    { // Complemented Cumulative Distribution Function beta.

      BOOST_MATH_STD_USING // for ADL of std functions

      static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";

      RealType const& x = c.param;
      beta_distribution<RealType, Policy> const& dist = c.dist;
      RealType a = dist.alpha();
      RealType b = dist.beta();

      // Argument checks:
      RealType result;
      if(false == beta_detail::check_dist_and_x(
        function,
        a, b, x,
        &result, Policy()))
      {
        return result;
      }
      if (x == 0)
      {
        return 1;
      }
      else if (x == 1)
      {
        return 0;
      }
      // Calculate cdf beta using the incomplete beta function.
      // Use of ibeta here prevents cancellation errors in calculating
      // 1 - x if x is very small, perhaps smaller than machine epsilon.
      return ibetac(a, b, x, Policy());
    } // beta cdf

    template <class RealType, class Policy>
    inline RealType quantile(const beta_distribution<RealType, Policy>& dist, const RealType& p)
    { // Quantile or Percent Point beta function or
      // Inverse Cumulative probability distribution function CDF.
      // Return x (0 <= x <= 1),
      // for a given probability p (0 <= p <= 1).
      // These functions take a probability as an argument
      // and return a value such that the probability that a random variable x
      // will be less than or equal to that value
      // is whatever probability you supplied as an argument.

      static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";

      RealType result; // of argument checks:
      RealType a = dist.alpha();
      RealType b = dist.beta();
      if(false == beta_detail::check_dist_and_prob(
        function,
        a, b, p,
        &result, Policy()))
      {
        return result;
      }
      // Special cases:
      if (p == 0)
      {
        return 0;
      }
      if (p == 1)
      {
        return 1;
      }
      return ibeta_inv(a, b, p, static_cast<RealType*>(0), Policy());
    } // quantile

    template <class RealType, class Policy>
    inline RealType quantile(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
    { // Complement Quantile or Percent Point beta function .
      // Return the number of expected x for a given
      // complement of the probability q.

      static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";

      //
      // Error checks:
      RealType q = c.param;
      const beta_distribution<RealType, Policy>& dist = c.dist;
      RealType result;
      RealType a = dist.alpha();
      RealType b = dist.beta();
      if(false == beta_detail::check_dist_and_prob(
        function,
        a,
        b,
        q,
        &result, Policy()))
      {
        return result;
      }
      // Special cases:
      if(q == 1)
      {
        return 0;
      }
      if(q == 0)
      {
        return 1;
      }

      return ibetac_inv(a, b, q, static_cast<RealType*>(0), Policy());
    } // 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_DIST_BETA_HPP