.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/continuous/plot_01_gamma_generalizes_exponential.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_probability_continuous_plot_01_gamma_generalizes_exponential.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_continuous_plot_01_gamma_generalizes_exponential.py:


The gamma distribution generalizes the exponential
==========================================================

A Gamma(shape=1, rate) distribution is exactly an Exponential(rate);
increasing the shape parameter builds up a sum of exponentials.

.. GENERATED FROM PYTHON SOURCE LINES 10-15

.. code-block:: Python

    import numpy as np

    from mathematicskit.probability import Exponential, Gamma, Normal
    from mathematicskit.probability.visualizers.plots import plot_distribution








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Gamma(1, rate) matches Exponential(rate) exactly
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-27

.. code-block:: Python


    rate = 2.0
    gamma = Gamma(shape=1.0, rate=rate)
    exponential = Exponential(rate=rate)
    xs = np.linspace(0.01, 3.0, 50)
    print("max |gamma(1,r) - exponential(r)| pdf difference:", np.max(np.abs(gamma.pdf(xs) - exponential.pdf(xs))))

    plot_distribution(Gamma(shape=5.0, rate=2.0))




.. image-sg:: /api/gallery/probability/continuous/images/sphx_glr_plot_01_gamma_generalizes_exponential_001.png
   :alt: Gamma
   :srcset: /api/gallery/probability/continuous/images/sphx_glr_plot_01_gamma_generalizes_exponential_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    max |gamma(1,r) - exponential(r)| pdf difference: 0.0

    <Axes: title={'center': 'Gamma'}, xlabel='x', ylabel='f(x)'>



.. GENERATED FROM PYTHON SOURCE LINES 28-30

The normal distribution and its MGF
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-34

.. code-block:: Python


    normal = Normal(mu=0.0, sigma=1.0)
    print("normal MGF at t=1:", normal.mgf(1.0), "expected exp(0.5):", np.exp(0.5))
    plot_distribution(normal)



.. image-sg:: /api/gallery/probability/continuous/images/sphx_glr_plot_01_gamma_generalizes_exponential_002.png
   :alt: Normal
   :srcset: /api/gallery/probability/continuous/images/sphx_glr_plot_01_gamma_generalizes_exponential_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    normal MGF at t=1: 1.6487212707001282 expected exp(0.5): 1.6487212707001282

    <Axes: title={'center': 'Normal'}, xlabel='x', ylabel='f(x)'>




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

   **Total running time of the script:** (0 minutes 0.046 seconds)


.. _sphx_glr_download_api_gallery_probability_continuous_plot_01_gamma_generalizes_exponential.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_gamma_generalizes_exponential.ipynb <plot_01_gamma_generalizes_exponential.ipynb>`

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

      :download:`Download Python source code: plot_01_gamma_generalizes_exponential.py <plot_01_gamma_generalizes_exponential.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_gamma_generalizes_exponential.zip <plot_01_gamma_generalizes_exponential.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_