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

.. _onemath_rng_poisson:

poisson
=======

Class is used for generation of Poisson distributed integer types random numbers.

.. _onemath_rng_poisson_description:

.. rubric:: Description

The class object is used in the :ref:`oneapi::math::rng::generate()<onemath_rng_generate>` function to provide random numbers Poisson distributed with distribution parameter :math:`\lambda`, where :math:`\lambda \in R; \lambda > 0;`.

The probability distribution is given by:

.. math::

    P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!}.

The cumulative distribution function is as follows:

.. math::

    F_{\lambda}(x) = \left\{ \begin{array}{rcl} \sum_{k = 0}^{\lfloor x \rfloor} \frac{\lambda^ke^{-\lambda}}{k!}, x \ge 0 \\ 0, x < 0 \end{array}\right.

.. _onemath_rng_poisson_syntax:

class poisson
-------------

.. rubric:: Syntax

.. code-block:: cpp

    namespace oneapi::math::rng {
    template<typename IntType = std::int32_t, typename Method = poisson_method::by_default>
    class poisson {
    public:
        using method_type = Method;
        using result_type = IntType;
        poisson();
        explicit poisson(double lambda);
        double lambda() const;
    };
    }

.. container:: section

    .. rubric:: Template parameters

    .. container:: section

        typename IntType
            Type of the produced values. Supported types:
                * ``std::int32_t``

    .. container:: section

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

                * ``oneapi::math::rng::poisson_method::by_default``
                * ``oneapi::math::rng::poisson_method::ptpe``
                * ``oneapi::math::rng::poisson_method::gaussian_icdf_based``

            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
        * - `poisson()`_
          - Default constructor
        * - `explicit poisson(double lambda)`_
          - Constructor with parameters
        * - `double lambda() const`_
          - Method to obtain distribution parameter

.. container:: section

    .. rubric:: Member types

    .. container:: section

        .. code-block:: cpp

            poisson::method_type = Method

        .. container:: section

            .. rubric:: Description

            The type which defines transformation method for generation.

    .. container:: section

        .. code-block:: cpp

            poisson::result_type = IntType

        .. container:: section

            .. rubric:: Description

            The type which defines type of generated random numbers.

.. container:: section

    .. rubric:: Constructors

    .. container:: section

        .. _`poisson()`:

        .. code-block:: cpp

            poisson::poisson()

        .. container:: section

            .. rubric:: Description

            Default constructor for distribution, parameters set as `lambda` = 0.5.

    .. container:: section

        .. _`explicit poisson(double lambda)`:

        .. code-block:: cpp

            explicit poisson::poisson(double lambda)

        .. container:: section

            .. rubric:: Description

            Constructor with parameters. `lambda` is a distribution parameter.

        .. container:: section

            .. rubric:: Throws

            oneapi::math::invalid_argument
                Exception is thrown when :math:`lambda \leq 0.0`

.. container:: section

    .. rubric:: Characteristics

    .. container:: section

        .. _`double lambda() const`:

        .. code-block:: cpp

            double poisson::lambda() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter `lambda`.

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