.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/limit_theorems/plot_03_chebyshev_inequality.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_limit_theorems_plot_03_chebyshev_inequality.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_limit_theorems_plot_03_chebyshev_inequality.py:


Chebyshev's inequality: one bound for every distribution
========================================================

:math:`P(|X-\mu| \geq k\sigma) \leq 1/k^2` holds for any distribution
with finite variance. Comparing it with exact tails shows how
conservative a universal bound has to be.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import Exponential, Normal, Poisson, Uniform, chebyshev_tail








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Exact tails against the bound
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-33

.. code-block:: Python


    k = np.linspace(1.0, 5.0, 200)
    fig, ax = plt.subplots()
    ax.semilogy(k, 1.0 / k**2, "k--", lw=2, label=r"Chebyshev $1/k^2$")
    for name, dist in [("Normal", Normal(0.0, 1.0)), ("Exponential", Exponential(rate=1.0)), ("Uniform", Uniform(0.0, 1.0)), ("Poisson(4)", Poisson(mu=4.0))]:
        result = chebyshev_tail(dist, k)
        ax.semilogy(k, np.maximum(result.exact, 1e-8), label=name)
    ax.set_ylim(1e-7, 1.5)
    ax.set_xlabel(r"$k$ (standard deviations)")
    ax.set_ylabel(r"$P(|X-\mu| \geq k\sigma)$")
    ax.legend()
    ax.set_title("Bienaymé-Chebyshev inequality")

    print(chebyshev_tail(Normal(0.0, 1.0), [2.0, 3.0]))



.. image-sg:: /api/gallery/probability/limit_theorems/images/sphx_glr_plot_03_chebyshev_inequality_001.png
   :alt: Bienaymé-Chebyshev inequality
   :srcset: /api/gallery/probability/limit_theorems/images/sphx_glr_plot_03_chebyshev_inequality_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    TailBoundResult(k=array([2., 3.]), bound=array([0.25      , 0.11111111]), exact=array([0.04550026, 0.0026998 ]))





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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_probability_limit_theorems_plot_03_chebyshev_inequality.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_chebyshev_inequality.ipynb <plot_03_chebyshev_inequality.ipynb>`

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

      :download:`Download Python source code: plot_03_chebyshev_inequality.py <plot_03_chebyshev_inequality.py>`

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

      :download:`Download zipped: plot_03_chebyshev_inequality.zip <plot_03_chebyshev_inequality.zip>`


.. only:: html

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

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