.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/programmers/plot_distributions.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_programmers_plot_distributions.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_programmers_plot_distributions.py:


biogeme.distributions
=====================

Example of usage of the `distributions` module.  This is for programmers who need
examples of use of the functions of the class. The examples are
designed to illustrate the syntax.

Michel Bierlaire
Sun Jun 29 2025, 07:12:12

.. GENERATED FROM PYTHON SOURCE LINES 13-28

.. code-block:: Python


    from IPython.core.display_functions import display

    from biogeme.distributions import (
        logisticcdf,
        lognormalpdf,
        normalpdf,
        triangularpdf,
        uniformpdf,
    )
    from biogeme.expressions import Beta
    from biogeme.version import get_text

    print(get_text())





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

 .. code-block:: none

    biogeme 3.3.3a0 [2025-12-25]
    Home page: http://biogeme.epfl.ch
    Submit questions to https://groups.google.com/d/forum/biogeme
    Michel Bierlaire, Transport and Mobility Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL)





.. GENERATED FROM PYTHON SOURCE LINES 29-34

pdf of the normal distributio: returns the biogeme expression of the
probability density function of the normal distribution:

.. math:: f(x;\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}}
          \exp{-\frac{(x-\mu)^2}{2\sigma^2}}.

.. GENERATED FROM PYTHON SOURCE LINES 36-37

Calculated for a numeric value.

.. GENERATED FROM PYTHON SOURCE LINES 37-40

.. code-block:: Python

    resulting_expression = normalpdf(0)
    display(resulting_expression)





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

 .. code-block:: none

    (exp((((-(`0.0` - `0.0`)) * (`0.0` - `0.0`)) / ((`2.0` * `1.0`) * `1.0`))) / (`1.0` * `2.506628275`))




.. GENERATED FROM PYTHON SOURCE LINES 41-43

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    np.float64(0.39894228034270457)



.. GENERATED FROM PYTHON SOURCE LINES 44-45

Calculated for an expression.

.. GENERATED FROM PYTHON SOURCE LINES 45-49

.. code-block:: Python

    a_parameter = Beta('a_parameter', 0, None, None, 1)
    mu = Beta('mu', 0, None, None, 1)
    sigma = Beta('sigma', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 50-53

.. code-block:: Python

    resulting_expression = normalpdf(a_parameter, mu=mu, s=sigma)
    display(resulting_expression)





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

 .. code-block:: none

    (exp((((-(Beta('a_parameter', 0, None, None, 1) - Beta('mu', 0, None, None, 1))) * (Beta('a_parameter', 0, None, None, 1) - Beta('mu', 0, None, None, 1))) / ((`2.0` * Beta('sigma', 1, None, None, 1)) * Beta('sigma', 1, None, None, 1)))) / (Beta('sigma', 1, None, None, 1) * `2.506628275`))




.. GENERATED FROM PYTHON SOURCE LINES 54-56

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    np.float64(0.39894228034270457)



.. GENERATED FROM PYTHON SOURCE LINES 57-62

pdf of the lognormal distribution: returns the biogeme expression of
the probability density function of the lognormal distribution

.. math:: f(x;\mu, \sigma) = \frac{1}{x\sigma \sqrt{2\pi}}
              \exp{-\frac{(\ln x-\mu)^2}{2\sigma^2}}

.. GENERATED FROM PYTHON SOURCE LINES 64-65

Calculated for a numeric value.

.. GENERATED FROM PYTHON SOURCE LINES 65-68

.. code-block:: Python

    resulting_expression = lognormalpdf(1)
    display(resulting_expression)





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

 .. code-block:: none

    (((`1.0` > `0.0`) * exp((((-(log(`1.0`) - `0.0`)) * (log(`1.0`) - `0.0`)) / ((`2.0` * `1.0`) * `1.0`)))) / ((`1.0` * `1.0`) * `2.506628275`))




.. GENERATED FROM PYTHON SOURCE LINES 69-71

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    np.float64(0.39894228034270457)



.. GENERATED FROM PYTHON SOURCE LINES 72-73

Calculated for an expression.

.. GENERATED FROM PYTHON SOURCE LINES 73-77

.. code-block:: Python

    a_parameter = Beta('a_parameter', 1, None, None, 1)
    mu = Beta('mu', 0, None, None, 1)
    sigma = Beta('sigma', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 78-81

.. code-block:: Python

    resulting_expression = lognormalpdf(a_parameter, mu=mu, s=sigma)
    display(resulting_expression)





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

 .. code-block:: none

    (((Beta('a_parameter', 1, None, None, 1) > `0.0`) * exp((((-(log(Beta('a_parameter', 1, None, None, 1)) - Beta('mu', 0, None, None, 1))) * (log(Beta('a_parameter', 1, None, None, 1)) - Beta('mu', 0, None, None, 1))) / ((`2.0` * Beta('sigma', 1, None, None, 1)) * Beta('sigma', 1, None, None, 1))))) / ((Beta('a_parameter', 1, None, None, 1) * Beta('sigma', 1, None, None, 1)) * `2.506628275`))




.. GENERATED FROM PYTHON SOURCE LINES 82-84

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    np.float64(0.39894228034270457)



.. GENERATED FROM PYTHON SOURCE LINES 85-91

pdf of the uniform distribution: returns the biogeme expression of
the probability density function of the uniform distribution

.. math:: f(x; a, b) = \left\{ \begin{array}{ll}
              \frac{1}{b-a} & \mbox{for } x \in [a, b] \\
              0 & \mbox{otherwise}\end{array} \right.

.. GENERATED FROM PYTHON SOURCE LINES 93-94

Calculated for a numeric value

.. GENERATED FROM PYTHON SOURCE LINES 94-97

.. code-block:: Python

    resulting_expression = uniformpdf(0)
    display(resulting_expression)





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

 .. code-block:: none

    ((((`0.0` < `-1.0`) * `0.0`) + ((`0.0` > `1.0`) * `0.0`)) + (((`0.0` >= `-1.0`) * (`0.0` <= `1.0`)) / (`1.0` - `-1.0`)))




.. GENERATED FROM PYTHON SOURCE LINES 98-100

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    0.5



.. GENERATED FROM PYTHON SOURCE LINES 101-102

Calculated for an expression

.. GENERATED FROM PYTHON SOURCE LINES 102-106

.. code-block:: Python

    a_parameter = Beta('a_parameter', 0, None, None, 1)
    a = Beta('a', -1, None, None, 1)
    b = Beta('b', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 107-110

.. code-block:: Python

    resulting_expression = uniformpdf(a_parameter, a=a, b=b)
    display(resulting_expression)





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

 .. code-block:: none

    ((((Beta('a_parameter', 0, None, None, 1) < Beta('a', -1, None, None, 1)) * `0.0`) + ((Beta('a_parameter', 0, None, None, 1) > Beta('b', 1, None, None, 1)) * `0.0`)) + (((Beta('a_parameter', 0, None, None, 1) >= Beta('a', -1, None, None, 1)) * (Beta('a_parameter', 0, None, None, 1) <= Beta('b', 1, None, None, 1))) / (Beta('b', 1, None, None, 1) - Beta('a', -1, None, None, 1))))




.. GENERATED FROM PYTHON SOURCE LINES 111-113

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    0.5



.. GENERATED FROM PYTHON SOURCE LINES 114-125

pdf of the triangular distribution: returns the biogeme expression
of the probability density function of the triangular distribution


.. math:: f(x;a, b, c) = \left\{ \begin{array}{ll} 0 &
             \text{if } x < a \\\frac{2(x-a)}{(b-a)(c-a)} &
             \text{if } a \leq x < c \\\frac{2(b-x)}{(b-a)(b-c)} &
             \text{if } c \leq x < b \\0 & \text{if } x \geq b.
             \end{array} \right.

It is assumed that :math:`a < c < b`. It is not verified.

.. GENERATED FROM PYTHON SOURCE LINES 127-128

Calculated for a numeric value.

.. GENERATED FROM PYTHON SOURCE LINES 128-131

.. code-block:: Python

    resulting_expression = triangularpdf(0)
    display(resulting_expression)





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

 .. code-block:: none

    MultipleSum(((`0.0` < `-1.0`) * `0.0`), ((((`0.0` >= `-1.0`) * (`0.0` < `0.0`)) * `2.0`) * ((`0.0` - `-1.0`) / ((`1.0` - `-1.0`) * (`0.0` - `-1.0`)))), (((`0.0` == `0.0`) * `2.0`) / (`1.0` - `-1.0`)), (((((`0.0` > `0.0`) * (`0.0` <= `1.0`)) * `2.0`) * (`1.0` - `0.0`)) / ((`1.0` - `-1.0`) * (`1.0` - `0.0`))), ((`0.0` > `1.0`) * `0.0`))




.. GENERATED FROM PYTHON SOURCE LINES 132-134

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    1.0



.. GENERATED FROM PYTHON SOURCE LINES 135-136

Calculated for an expression.

.. GENERATED FROM PYTHON SOURCE LINES 136-141

.. code-block:: Python

    a_parameter = Beta('a_parameter', 0, None, None, 1)
    a = Beta('a', -1, None, None, 1)
    b = Beta('b', 1, None, None, 1)
    c = Beta('c', 0, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 142-145

.. code-block:: Python

    resulting_expression = triangularpdf(a_parameter, a=a, b=b, c=c)
    display(resulting_expression)





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

 .. code-block:: none

    MultipleSum(((Beta('a_parameter', 0, None, None, 1) < Beta('a', -1, None, None, 1)) * `0.0`), ((((Beta('a_parameter', 0, None, None, 1) >= Beta('a', -1, None, None, 1)) * (Beta('a_parameter', 0, None, None, 1) < Beta('c', 0, None, None, 1))) * `2.0`) * ((Beta('a_parameter', 0, None, None, 1) - Beta('a', -1, None, None, 1)) / ((Beta('b', 1, None, None, 1) - Beta('a', -1, None, None, 1)) * (Beta('c', 0, None, None, 1) - Beta('a', -1, None, None, 1))))), (((Beta('a_parameter', 0, None, None, 1) == Beta('c', 0, None, None, 1)) * `2.0`) / (Beta('b', 1, None, None, 1) - Beta('a', -1, None, None, 1))), (((((Beta('a_parameter', 0, None, None, 1) > Beta('c', 0, None, None, 1)) * (Beta('a_parameter', 0, None, None, 1) <= Beta('b', 1, None, None, 1))) * `2.0`) * (Beta('b', 1, None, None, 1) - Beta('a_parameter', 0, None, None, 1))) / ((Beta('b', 1, None, None, 1) - Beta('a', -1, None, None, 1)) * (Beta('b', 1, None, None, 1) - Beta('c', 0, None, None, 1)))), ((Beta('a_parameter', 0, None, None, 1) > Beta('b', 1, None, None, 1)) * `0.0`))




.. GENERATED FROM PYTHON SOURCE LINES 146-148

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    1.0



.. GENERATED FROM PYTHON SOURCE LINES 149-156

CDF of the logistic distribution: returns the biogeme expression of
the cumulative distribution function of the logistic distribution


.. math:: f(x;\mu, \sigma) = \frac{1}{1+\exp\left(-\frac{x-\mu}{\sigma} \right)}



.. GENERATED FROM PYTHON SOURCE LINES 158-159

Calculated for a numeric value

.. GENERATED FROM PYTHON SOURCE LINES 159-162

.. code-block:: Python

    resulting_expression = logisticcdf(0)
    display(resulting_expression)





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

 .. code-block:: none

    (`1.0` / (`1.0` + exp(((-(`0.0` - `0.0`)) / `1.0`))))




.. GENERATED FROM PYTHON SOURCE LINES 163-165

.. code-block:: Python

    resulting_expression.get_value()





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

 .. code-block:: none


    np.float64(0.5)



.. GENERATED FROM PYTHON SOURCE LINES 166-167

Calculated for an expression

.. GENERATED FROM PYTHON SOURCE LINES 167-171

.. code-block:: Python

    a_parameter = Beta('a_parameter', 0, None, None, 1)
    mu = Beta('mu', 0, None, None, 1)
    sigma = Beta('sigma', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 172-175

.. code-block:: Python

    resulting_expression = logisticcdf(a_parameter, mu=mu, s=sigma)
    display(resulting_expression)





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

 .. code-block:: none

    (`1.0` / (`1.0` + exp(((-(Beta('a_parameter', 0, None, None, 1) - Beta('mu', 0, None, None, 1))) / Beta('sigma', 1, None, None, 1)))))




.. GENERATED FROM PYTHON SOURCE LINES 176-177

.. code-block:: Python

    resulting_expression.get_value()




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

 .. code-block:: none


    np.float64(0.5)




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

   **Total running time of the script:** (0 minutes 0.003 seconds)


.. _sphx_glr_download_auto_examples_programmers_plot_distributions.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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