.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_probabilistic_modeling/distributions/plot_create_your_own_dist.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_probabilistic_modeling_distributions_plot_create_your_own_dist.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_probabilistic_modeling_distributions_plot_create_your_own_dist.py:


Create your own distribution given its quantile function
========================================================

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We want to create a distribution with CDF :math:`F` from the quantile function :math:`F^{-1}`.
In order to implement this, we use the :class:`~openturns.CompositeDistribution` class.

.. GENERATED FROM PYTHON SOURCE LINES 12-37

We know that the random variable :math:`X` is distributed according to :math:`F`
if and only if :math:`U=F(X)` is distributed according to the uniform
distribution in the :math:`[0,1]` interval, i.e. :math:`U=F(X) \sim \mathcal{U}(0,1)`.
Hence, if :math:`U \sim  \mathcal{U}(0,1)` then :math:`X=F^{-1}(U)` is distributed according to :math:`F`.

In this example, we want to create a distribution with CDF :math:`F: \mathbb{R} \rightarrow [0,1] `parametrized by :math:`\rho > 1`:

.. math::
   F(x) = 1-e^{-\rho^x} \quad \forall x  \in \mathbb{R}.


The quantile function is :math:`F^{-1} : u \rightarrow [0,1]` and writes:

.. math::
   F^{-1}(u) = \dfrac{\log \left[ - \log (1-u) \right] }{\log(\rho)} \quad \forall u  \in [0,1]


Since :math:`U \sim \mathcal{U}(0,1)`, then :math:`(1-U)\sim\mathcal{U}(0,1)`.
This is why we can simplify the expression and define the function :math:`G` such as:

.. math::
   G(u) = \dfrac{\log \left[ - \log u \right] }{\log(\rho)} \quad \forall u  \in [0,1].


Then :math:`G(U)` is distributed according to the :math:`F` distribution.

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First, we import the useful libraries and we create the symbolic function :math:`G`.

.. GENERATED FROM PYTHON SOURCE LINES 42-45

.. code-block:: Python

    import openturns as ot
    from openturns.viewer import View








.. GENERATED FROM PYTHON SOURCE LINES 46-49

Then, we create the :math:`G` function with :math:`\rho = 2.0`.
To do this, we create a function which takes both :math:`y` and :math:`\rho` as inputs and returns :math:`G(u)`.
Then the `g` function is defined as a :class:`~openturns.ParametricFunction` with a fixed value of :math:`\rho`.

.. GENERATED FROM PYTHON SOURCE LINES 51-55

.. code-block:: Python

    gWithParameter = ot.SymbolicFunction(["u", "rho"], ["log(-log(u)) / log(rho)"])
    rho = 2.0
    g = ot.ParametricFunction(gWithParameter, [1], [rho])








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We define the distribution distF as the image through :math:`G` of the Uniform(0,1) distribution:

.. GENERATED FROM PYTHON SOURCE LINES 59-61

.. code-block:: Python

    distF = ot.CompositeDistribution(g, ot.Uniform(0.0, 1.0))








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Now, we can draw its pdf, cdf, sample it,...

.. GENERATED FROM PYTHON SOURCE LINES 65-70

.. code-block:: Python

    g = distF.drawPDF()
    g.setTitle("A distribution based on the quantile function.")
    g.setLegendPosition("")
    view = View(g)
    view.ShowAll()



.. image-sg:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_create_your_own_dist_001.png
   :alt: A distribution based on the quantile function.
   :srcset: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_create_your_own_dist_001.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_create_your_own_dist.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_create_your_own_dist.ipynb <plot_create_your_own_dist.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_create_your_own_dist.py <plot_create_your_own_dist.py>`