.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/lifetimes/plot_02_poisson_failures.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_reliability_lifetimes_plot_02_poisson_failures.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_reliability_lifetimes_plot_02_poisson_failures.py:


Poisson's law of rare failures (1837)
=====================================

Poisson (1837) derived the distribution of the number of rare events among
many trials, :math:`P(k) = (\lambda T)^k e^{-\lambda T}/k!`
(:func:`~engineeringkit.reliability.poisson_failures_probability`). A fleet
of units failing independently at a constant rate :math:`\lambda` has
:math:`\lambda T` expected failures in total operating time :math:`T`
(:func:`~engineeringkit.reliability.expected_failures`), and the count is
Poisson. The example simulates failures as exponential gaps and checks the
counts, then sizes a spares kit.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.reliability import expected_failures, poisson_failures_probability








.. GENERATED FROM PYTHON SOURCE LINES 22-27

Simulate failure counts
-----------------------

50 units, each failing at :math:`2\times10^{-5}` per hour, run for a year
of 8760 h; every failed unit is replaced at once. Time unit: hours.

.. GENERATED FROM PYTHON SOURCE LINES 27-41

.. code-block:: Python


    rng = np.random.default_rng(1837)
    lam, units, hours = 2e-5, 50, 8760.0
    mean = expected_failures(lam, units * hours)
    trials = 20_000
    gaps = rng.exponential(1 / lam, size=(trials, units, 8))
    counts = (np.cumsum(gaps, axis=2) <= hours).sum(axis=(1, 2))
    k = np.arange(0, 25)
    pmf = np.array([poisson_failures_probability(int(i), lam, units * hours) for i in k])
    observed = np.bincount(counts, minlength=k.size)[: k.size] / trials
    print(f"expected failures per year: {mean:.2f}; simulated mean {counts.mean():.2f}")
    assert np.isclose(pmf.sum(), 1, atol=1e-6)
    assert np.abs(observed - pmf).max() < 0.01





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

 .. code-block:: none

    expected failures per year: 8.76; simulated mean 8.77




.. GENERATED FROM PYTHON SOURCE LINES 42-47

How many spares?
----------------

Stock enough spares that the year's failures exceed them with
probability under 1%.

.. GENERATED FROM PYTHON SOURCE LINES 47-62

.. code-block:: Python


    cumulative = np.cumsum(pmf)
    spares = int(np.argmax(cumulative >= 0.99))
    print(f"spares for 99% cover: {spares} (P = {cumulative[spares]:.4f})")
    fig, ax = plt.subplots(figsize=(6.5, 3.5))
    ax.bar(k, observed, alpha=0.5, label="simulated fleet")
    ax.plot(k, pmf, "o-", color="C1", label="Poisson")
    ax.axvline(spares + 0.5, color="gray", ls="--", label=f"{spares} spares")
    ax.set(xlabel="failures in one year", ylabel="probability")
    ax.legend()
    fig.tight_layout()

    assert cumulative[spares] >= 0.99 > cumulative[spares - 1]

    plt.show()



.. image-sg:: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_02_poisson_failures_001.png
   :alt: plot 02 poisson failures
   :srcset: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_02_poisson_failures_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    spares for 99% cover: 16 (P = 0.9913)





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

   **Total running time of the script:** (0 minutes 0.070 seconds)


.. _sphx_glr_download_api_gallery_reliability_lifetimes_plot_02_poisson_failures.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/reliability/lifetimes/plot_02_poisson_failures.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_poisson_failures.ipynb <plot_02_poisson_failures.ipynb>`

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

      :download:`Download Python source code: plot_02_poisson_failures.py <plot_02_poisson_failures.py>`

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

      :download:`Download zipped: plot_02_poisson_failures.zip <plot_02_poisson_failures.zip>`


.. only:: html

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

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