.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_distributions_plot_createYourOwnDist-vMB.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_createYourOwnDist-vMB.py:


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

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 `CompositeDistribution <http://openturns.github.io/openturns/latest/user_manual/_generated/openturns.CompositeDistribution.html#openturns.CompositeDistribution>`_ class.

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.

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


.. code-block:: default

    import openturns as ot
    from openturns.viewer import View








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 `ParametricFunction` with a fixed value of :math:`\rho`.


.. code-block:: default

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








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


.. code-block:: default

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








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


.. code-block:: default

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



.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_createYourOwnDist-vMB_001.png
    :alt: A distribution based on the quantile function.
    :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.074 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_createYourOwnDist-vMB.py:


.. only :: html

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

     :download:`Download Python source code: plot_createYourOwnDist-vMB.py <plot_createYourOwnDist-vMB.py>`



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     :download:`Download Jupyter notebook: plot_createYourOwnDist-vMB.ipynb <plot_createYourOwnDist-vMB.ipynb>`


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