.. _distributions-template-parameter-method:

Distributions Template Parameter Method
=======================================

.. tabularcolumns:: |\Y{0.4}|\Y{0.2}|\Y{0.4}|

.. list-table::
   :header-rows: 1
   :class: longtable

   * -  Method Type
     -  Distributions
     -  Math Description

   * -  ``uniform_method::standard``

        ``uniform_method::accurate``
     -  ``uniform``
     -   Standard method. Currently there is only one method for these functions. ``uniform_method::accurate`` checks for additional ``s`` and ``d`` data types. For ``integer`` data types, it uses ``d`` as a ``BRNG`` data type (``s`` ``BRNG`` data type is used in ``uniform_method::standard`` method on GPU).
   * -  ``gaussian_method::box_muller``
     -  ``gaussian``
     -  Generates normally distributed random number x thru the pair of uniformly distributed numbers u1 and u2 according to the formula: \ :math:`x = \sqrt {-2 \ln u_1} \sin {2 \pi u_2}`\
   * -  ``gaussian_method::box_muller2``
     -  ``gaussian``
     -  Generates normally distributed random numbers x1 and x2 thru the pair of uniformly distributed numbers u1 and u2 according to   the formulas: \ :math:`x_1 = \sqrt{-2 \ln u_1} \sin {2 \pi u_2}`\       \ :math:`x_2 = \sqrt{-2 \ln u_1} \cos {2 \pi u_2}`\       Lognormal   distribution: generated normally distributed random numbers x1 and x2   are converted to lognormal distribution.
   * -  ``gaussian_method::icdf``
        ``geometric_method::icdf``
     -  ``gaussian``

        ``geometric``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``exponential_method::icdf``
        ``exponential_method::icdf_accurate``
     -  ``exponential``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``weibull_method::icdf``

        ``weibull_method::icdf_accurate``
     -  ``weibull``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``cauchy_method::icdf``
     -  ``cauchy``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``rayleigh_method::icdf``

        ``rayleigh_method::icdf_accurate``
     -  ``rayleigh``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``lognormal_method::icdf``
        ``lognormal_method::icdf_accurate``
     -  ``lognormal``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``lognormal_method::box_muller2``
        ``lognormal_method::box_muller2_accurate``
     -  ``lognormal``
     -  Normally distributed random numbers x1 and x2 are produced through the pair of uniformly distributed numbers u1 and u2 according to the formulas: \ :math:`x_1 = -2 \ln u_1 \sin {2 \pi u_2}`\       \ :math:`x_2 = -2 \ln u_1 \cos {2 \pi u_2}`\   Then x1 and x2 are converted to lognormal distribution.
   * -  ``gumbel_method::icdf``
     -  ``gumbel``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``bernoulli_method::icdf``
     -  ``bernoulli``
     -  Inverse cumulative distribution function (ICDF) method.
   * -  ``gamma_method::marsaglia``
        ``gamma_method::marsaglia_accurate``
     -  ``gamma``
     -  For ``α > 1``, a gamma distributed random number is generated as a cube of properly scaled normal random number; for ``0.6 ≤α < 1``, a gamma distributed random number is generated using rejection from Weibull distribution; for ``α < 0.6``, a gamma distributed random number is obtained using transformation of exponential power distribution; for ``α = 1``, gamma distribution is reduced to exponential distribution.
   * -  ``beta_method::cja``

        ``beta_method::cja_accurate``
     -  ``beta``
     -  Cheng-Johnk-Atkinson method. For ``min(p, q) > 1``, Cheng method is used; for ``min(p, q) < 1``, Johnk method is used, if ``q + K·p2+   C≤ 0 (K = 0.852..., C=-0.956...)`` otherwise, Atkinson switching algorithm is used; for ``max(p, q) < 1``, method of Johnk is used; for ``min(p, q) < 1, max(p, q)> 1``, Atkinson switching algorithm is used (CJA stands for Cheng, Johnk, Atkinson); for ``p = 1or q = 1``, inverse cumulative distribution function method is used; for ``p = 1`` and ``q = 1``, beta distribution is reduced to uniform distribution.
   * -  ``chi_square_method::gamma_based``
     -  ``chi_square``
     -  (most common): If ``ν ≥ 17`` or ν is odd and ``5 ≤ ν ≤ 15``, a chi-square distribution is reduced to a Gamma distribution with these parameters: Shape ``α = ν / 2`` Offset a = 0 Scale factor ``β = 2``. The random numbers of the Gamma distribution are generated.
   * -  ``gaussian_mv_method::box_muller``
        ``gaussian_mv_method::box_muller2``
        ``gaussian_mv_method::icdf``
     -  ``gaussian_mv``
     -  BoxMuller method for multivariate Gaussian distribution. BoxMuller_2 method for multivariate Gaussian distribution. Inverse cumulative distribution   function (ICDF) method.
   * -  ``binomial_method::btpe``
     -  ``binomial``
     -  Acceptance/rejection method for ``ntrial·min(p, 1p) ≥ 30`` with decomposition into four regions:

        Two parallelograms

        Triangle

        Left exponential tail

        Right exponenetial tail

   * -  ``poisson_method::ptpe``
     -  ``poisson``
     -  Acceptance/rejection method for ``λ≥ 27`` with decomposition into four regions:

        Two parallelograms

        Triangle

        Left exponential tail

        Right exponenetial tail
   * -  ``poisson_method::gaussian_icdf_based``
        ``poisson_v_method::gaussian_icdf_based``
     -  ``poisson``
        ``poisson_v``
     -  for ``λ≥ 1``, method based on Poisson inverse CDF approximation by Gaussian inverse CDF; for ``λ < 1``, table lookup method is used.
   * -  ``hypergeometric_method::h2pe``
     -  ``hypergeometric``
     -  Acceptance/rejection method for large mode of distribution with decomposition into three regions:

        Rectangular

        Left exponential tail

        Right exponential tail
   * -  ``negative_binomial_method::nbar``
     -  ``negative_binomial``
     -  Acceptance/rejection method for: \ :math:`\frac{(a-1) \cdot (1-p)}{p} \geq 100`\ with decomposition into five regions:

        Rectangular

        (\2) trapezoid

        Left exponential tail

        Right exponential tail

   * -  ``multinomial_method::poisson_icdf_based``
     -  ``multinomial``
     -  Multinomial distribution with parameters ``m``, ``k``, and a probability vector ``p``. Random numbers of the multinomial distribution are generated by Poisson Approximation method.