.. SPDX-FileCopyrightText: 2019-2020 Intel Corporation
..
.. SPDX-License-Identifier: CC-BY-4.0

.. _onemath_rng_beta:

beta
====

Class is used for generation of beta distributed real types random numbers.

.. _onemath_rng_beta_description:

.. rubric:: Description

The class object is used in the :ref:`oneapi::math::rng::generate()<onemath_rng_generate>`
function to provide random numbers beta distributed with shape parameters :math:`p` and :math:`q`,
displacement :math:`\alpha` and scale parameter :math:`(b, \beta)`, where :math:`p`, :math:`q`.
:math:`\alpha`, :math:`\beta` :math:`\in R; p > 0; q > 0; \beta > 0`.

The probability distribution is given by:

.. math::

    f_{p, q, \alpha, \beta}(x) = \left\{ \begin{array}{rcl} \frac{1}{B(p, q) * \beta^{p + q - 1}}(x - a)^{p - 1}*(\beta + \alpha - x)^{q - 1}, \alpha \leq x < \alpha + \beta \\ 0, x < \alpha, x \ge \alpha + \beta \end{array}\right.

The cumulative distribution function is as follows:

.. math::

    F_{a, b}(x) = \left\{ \begin{array}{rcl} 0, x < \alpha \\ \int^x_{\alpha}\frac{1}{B(p, q) * \beta^{p + q - 1}}(y - \alpha)^{p - 1}*(\beta + \alpha - y)^{q - 1}dy, \alpha \leq x < \alpha + \beta, x \in R \\ 1, x \ge \alpha + \beta \end{array}\right.


Where :math:`B(p, 1)` is the complete beta function.

.. _onemath_rng_beta_syntax:

class beta
----------

.. rubric:: Syntax

.. code-block:: cpp

    namespace oneapi::math::rng {
    template<typename RealType = float, typename Method = beta_method::by_default>
    class beta {
    public:
        using method_type = Method;
        using result_type = RealType;
        beta();
        explicit beta(RealType p, RealType q, RealType a, RealType b);
        RealType p() const;
        RealType q() const;
        RealType a() const;
        RealType b() const;
    };
    }

.. container:: section

    .. rubric:: Template parameters

    .. container:: section

        typename RealType
            Type of the produced values. Supported types:
                * ``float``
                * ``double``

    .. container:: section

        typename Method = oneapi::math::rng::beta_method::by_default
            Transformation method, which will be used for generation. Supported types:

                * ``oneapi::math::rng::beta_method::by_default``
                * ``oneapi::math::rng::beta_method::cja``
                * ``oneapi::math::rng::beta_method::cja_accurate``

            See description of the methods in :ref:`Distributions methods template parameter<onemath_rng_distributions_template_parameter_onemath_rng_method_values>`.

.. container:: section

    .. rubric:: Class Members

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

        * - Routine
          - Description
        * - `beta()`_
          - Default constructor
        * - `explicit beta(RealType p, RealType q, RealType a, RealType b)`_
          - Constructor with parameters
        * - `RealType p() const`_
          - Method to obtain shape ``p``
        * - `RealType q() const`_
          - Method to obtain shape ``q``
        * - `RealType a() const`_
          - Method to obtain displacement :math:`\alpha`
        * - `RealType b() const`_
          - Method to obtain scalefactor :math:`\beta`

.. container:: section

    .. rubric:: Member types

    .. container:: section

        .. code-block:: cpp

            beta::method_type = Method

        .. container:: section

            .. rubric:: Description

            The type which defines transformation method for generation.

    .. container:: section

        .. code-block:: cpp

            beta::result_type = RealType

        .. container:: section

            .. rubric:: Description

            The type which defines type of generated random numbers.

.. container:: section

    .. rubric:: Constructors

    .. container:: section

        .. _`beta()`:

        .. code-block:: cpp

            beta::beta()

        .. container:: section

            .. rubric:: Description

            Default constructor for distribution, parameters set as ``p`` = 1.0, ``q`` = 0.0, :math:`\alpha` = 1.0, :math:`\beta` = 1.0.

    .. container:: section

        .. _`explicit beta(RealType p, RealType q, RealType a, RealType b)`:

        .. code-block:: cpp

            explicit beta::beta(RealType p, RealType q, RealType a, RealType b)

        .. container:: section

            .. rubric:: Description

            Constructor with parameters. ``p`` and ``q`` are shapes, :math:`\alpha` is a displacement, :math:`\beta` is a scalefactor.

        .. container:: section

            .. rubric:: Throws

            oneapi::math::invalid_argument
                Exception is thrown when :math:`p \leq 0.0f`, or :math:`q \leq 0.0f`, or :math:`\beta \leq 0.0f`

.. container:: section

    .. rubric:: Characteristics

    .. container:: section

        .. _`RealType p() const`:

        .. code-block:: cpp

            RealType beta::p() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter ``p`` - shape.

    .. container:: section

        .. _`RealType q() const`:

        .. code-block:: cpp

            RealType beta::q() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter ``q`` - shape.

    .. container:: section

        .. _`RealType a() const`:

        .. code-block:: cpp

            RealType beta::a() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter :math:`\alpha` - displacement.

    .. container:: section

        .. _`RealType b() const`:

        .. code-block:: cpp

            RealType beta::b() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter :math:`\beta` - scalefactor.

**Parent topic:** :ref:`onemath_rng_distributions`