.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/basics/plot_02_box_muller.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_monte_carlo_basics_plot_02_box_muller.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_monte_carlo_basics_plot_02_box_muller.py:


Box-Muller normal variates
==========================

Every Black-Scholes simulation needs normal random numbers, and a
computer produces uniforms. George Box and Mervin Muller (1958) gave an
exact transform: if :math:`u_1, u_2` are independent uniforms,

.. math:: z_1 = \sqrt{-2\ln u_1}\cos(2\pi u_2), \qquad z_2 = \sqrt{-2\ln u_1}\sin(2\pi u_2)

are independent standard normals. The squared radius of a standard
bivariate normal is exponential with mean 2 and its angle is uniform, so
inverting both gives the pair.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import kstest, norm

    from financialkit.monte_carlo import box_muller, box_muller_normals

    rng = np.random.default_rng(0)
    u1, u2 = rng.random((2, 3000))
    z1, z2 = box_muller(1.0 - u1, u2)








.. GENERATED FROM PYTHON SOURCE LINES 28-32

Uniform square to normal plane
------------------------------
Colouring each uniform pair by its angle :math:`2\pi u_2` shows the map:
:math:`u_2` sets the direction and :math:`u_1` the radius.

.. GENERATED FROM PYTHON SOURCE LINES 32-40

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    axes[0].scatter(u1, u2, c=u2, s=3, cmap="twilight")
    axes[0].set_title("uniforms $(u_1, u_2)$")
    axes[1].scatter(z1, z2, c=u2, s=3, cmap="twilight")
    axes[1].set_aspect("equal")
    axes[1].set_title("Box-Muller $(z_1, z_2)$")




.. image-sg:: /api/gallery/monte_carlo/basics/images/sphx_glr_plot_02_box_muller_001.png
   :alt: uniforms $(u_1, u_2)$, Box-Muller $(z_1, z_2)$
   :srcset: /api/gallery/monte_carlo/basics/images/sphx_glr_plot_02_box_muller_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Box-Muller $(z_1, z_2)$')



.. GENERATED FROM PYTHON SOURCE LINES 41-43

The output is normal
--------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-53

.. code-block:: Python


    z = box_muller_normals(100_000, rng=1)
    x = np.linspace(-4, 4, 200)
    axes[2].hist(z, bins=80, density=True, alpha=0.6, label="Box-Muller")
    axes[2].plot(x, norm.pdf(x), "k", label="N(0, 1)")
    axes[2].legend()
    axes[2].set_title(f"KS p-value {kstest(z, 'norm').pvalue:.2f}")
    print(f"mean {z.mean():+.4f}, variance {z.var():.4f}, corr(z1, z2) {np.corrcoef(z1, z2)[0, 1]:+.4f}")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    mean +0.0029, variance 0.9990, corr(z1, z2) +0.0012





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

   **Total running time of the script:** (0 minutes 0.111 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_basics_plot_02_box_muller.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/monte_carlo/basics/plot_02_box_muller.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_box_muller.ipynb <plot_02_box_muller.ipynb>`

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

      :download:`Download Python source code: plot_02_box_muller.py <plot_02_box_muller.py>`

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

      :download:`Download zipped: plot_02_box_muller.zip <plot_02_box_muller.zip>`


.. only:: html

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

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