.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/branching/plot_01_galton_watson.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_branching_plot_01_galton_watson.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_branching_plot_01_galton_watson.py:


Galton-Watson: the probability a family name dies out
========================================================

Each man has 0, 1, or 2 sons with probabilities 0.2, 0.3, 0.5. The
extinction probability is the smallest fixed point of the generating
function :math:`G(s) = 0.2 + 0.3s + 0.5s^2`, namely :math:`q = 0.4`,
even though the population grows by 30% per generation on average.

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import galton_watson_extinction_probability, galton_watson_simulate

    pmf = [0.2, 0.3, 0.5]








.. GENERATED FROM PYTHON SOURCE LINES 20-22

The fixed point of the generating function
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-33

.. code-block:: Python


    q = galton_watson_extinction_probability(pmf)
    s = np.linspace(0.0, 1.0, 200)
    fig, ax = plt.subplots()
    ax.plot(s, np.polynomial.polynomial.polyval(s, pmf), label="G(s)")
    ax.plot(s, s, "k--", label="s")
    ax.plot(q, q, "o", color="tab:red", label=f"q = {q:.4f}")
    ax.set_xlabel("s")
    ax.legend()
    ax.set_title("Extinction probability = smallest root of G(s) = s")




.. image-sg:: /api/gallery/probability/branching/images/sphx_glr_plot_01_galton_watson_001.png
   :alt: Extinction probability = smallest root of G(s) = s
   :srcset: /api/gallery/probability/branching/images/sphx_glr_plot_01_galton_watson_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Extinction probability = smallest root of G(s) = s')



.. GENERATED FROM PYTHON SOURCE LINES 34-36

Simulated families
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-46

.. code-block:: Python


    result = galton_watson_simulate(pmf, n_generations=25, n_runs=10000, seed=0)
    print(f"simulated extinction fraction: {result.extinct_fraction:.4f} (theory {q:.4f})")

    fig, ax = plt.subplots()
    for sizes in result.generation_sizes[:40]:
        ax.semilogy(np.where(sizes > 0, sizes, np.nan), color="tab:blue", alpha=0.4)
    ax.set_xlabel("generation")
    ax.set_ylabel("population")
    ax.set_title("40 simulated families (dying lines stop)")



.. image-sg:: /api/gallery/probability/branching/images/sphx_glr_plot_01_galton_watson_002.png
   :alt: 40 simulated families (dying lines stop)
   :srcset: /api/gallery/probability/branching/images/sphx_glr_plot_01_galton_watson_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    simulated extinction fraction: 0.3951 (theory 0.4000)

    Text(0.5, 1.0, '40 simulated families (dying lines stop)')




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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_probability_branching_plot_01_galton_watson.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_galton_watson.ipynb <plot_01_galton_watson.ipynb>`

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

      :download:`Download Python source code: plot_01_galton_watson.py <plot_01_galton_watson.py>`

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

      :download:`Download zipped: plot_01_galton_watson.zip <plot_01_galton_watson.zip>`


.. only:: html

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

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