.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/discrete/plot_02_poisson_rare_events.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_discrete_plot_02_poisson_rare_events.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_discrete_plot_02_poisson_rare_events.py:


Poisson's law of rare events: the binomial limit
================================================

Many independent trials, each with a tiny chance of success, produce a
number of successes that follows a Poisson distribution. Poisson (1837)
obtained it as the limit of :math:`\mathrm{Binomial}(n, p)` with
:math:`n \to \infty`, :math:`p \to 0` and the mean :math:`np = \mu` held
fixed:

.. math::

   \binom{n}{k} p^k (1-p)^{n-k} \;\longrightarrow\; e^{-\mu}\frac{\mu^k}{k!}.

.. GENERATED FROM PYTHON SOURCE LINES 17-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import Binomial, Poisson

    mu = 4.0
    poisson = Poisson(mu=mu)
    ks = np.arange(0, 15)








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Binomial PMFs approach the Poisson PMF as events become rarer
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    fig, ax = plt.subplots()
    width = 0.25
    for offset, n in zip((-width, 0.0, width), (8, 20, 200)):
        ax.bar(ks + offset, Binomial(n=n, p=mu / n).pmf(ks), width=width, alpha=0.8, label=f"Binomial(n={n}, p={mu / n:.3g})")
    ax.plot(ks, poisson.pmf(ks), "ko-", label=rf"Poisson($\mu$={mu:g})")
    ax.set_xlabel("number of events $k$")
    ax.set_ylabel("P(X = k)")
    ax.set_title(r"Rare events: Binomial$(n, \mu/n)$ tends to Poisson$(\mu)$")
    ax.legend()




.. image-sg:: /api/gallery/probability/discrete/images/sphx_glr_plot_02_poisson_rare_events_001.png
   :alt: Rare events: Binomial$(n, \mu/n)$ tends to Poisson$(\mu)$
   :srcset: /api/gallery/probability/discrete/images/sphx_glr_plot_02_poisson_rare_events_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11a99cec0>



.. GENERATED FROM PYTHON SOURCE LINES 41-43

The largest PMF gap shrinks like 1/n
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-57

.. code-block:: Python


    ns = np.array([10, 30, 100, 300, 1000, 3000, 10000])
    gaps = np.array([np.max(np.abs(Binomial(n=int(n), p=mu / n).pmf(ks) - poisson.pmf(ks))) for n in ns])
    for n, gap in zip(ns, gaps):
        print(f"n={n:>6}: max |binomial - poisson| = {gap:.2e}")

    fig, ax = plt.subplots()
    ax.loglog(ns, gaps, "o-", label="max PMF difference")
    ax.loglog(ns, gaps[0] * ns[0] / ns, "k--", label=r"$\propto 1/n$")
    ax.set_xlabel("number of trials $n$ (with $np = 4$)")
    ax.set_ylabel("max |Binomial - Poisson|")
    ax.set_title("Poisson limit theorem")
    ax.legend()




.. image-sg:: /api/gallery/probability/discrete/images/sphx_glr_plot_02_poisson_rare_events_002.png
   :alt: Poisson limit theorem
   :srcset: /api/gallery/probability/discrete/images/sphx_glr_plot_02_poisson_rare_events_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n=    10: max |binomial - poisson| = 5.55e-02
    n=    30: max |binomial - poisson| = 1.44e-02
    n=   100: max |binomial - poisson| = 4.02e-03
    n=   300: max |binomial - poisson| = 1.31e-03
    n=  1000: max |binomial - poisson| = 3.92e-04
    n=  3000: max |binomial - poisson| = 1.30e-04
    n= 10000: max |binomial - poisson| = 3.91e-05

    <matplotlib.legend.Legend object at 0x11ce352b0>



.. GENERATED FROM PYTHON SOURCE LINES 58-63

A rare-event count: simulated arrivals
------------------------------------------------------------------

100000 independent units, each failing with probability 4e-5 in a
day: the daily count of failures is Poisson with mean 4.

.. GENERATED FROM PYTHON SOURCE LINES 63-67

.. code-block:: Python


    counts = Binomial(n=100000, p=4e-5).sample(size=20000, seed=0)
    print(f"mean daily failures = {counts.mean():.3f}, variance = {counts.var():.3f} (Poisson: both = {mu})")
    print(f"P(no failures): simulated {np.mean(counts == 0):.4f}, Poisson e^-4 = {poisson.pmf(0):.4f}")




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

 .. code-block:: none

    mean daily failures = 3.971, variance = 4.045 (Poisson: both = 4.0)
    P(no failures): simulated 0.0202, Poisson e^-4 = 0.0183





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

   **Total running time of the script:** (0 minutes 0.097 seconds)


.. _sphx_glr_download_api_gallery_probability_discrete_plot_02_poisson_rare_events.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_poisson_rare_events.ipynb <plot_02_poisson_rare_events.ipynb>`

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

      :download:`Download Python source code: plot_02_poisson_rare_events.py <plot_02_poisson_rare_events.py>`

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

      :download:`Download zipped: plot_02_poisson_rare_events.zip <plot_02_poisson_rare_events.zip>`


.. only:: html

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

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