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Loïc Guégan 2023-10-17 13:57:32 +02:00
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source/statistics/probability_distribution_functions.rst
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Probability Distribution Functions
------------------------------------
.. _PDF:
Probability Density Function
=============================
@ -43,4 +45,23 @@ Let's compute its CDF:
Cumulative Distribution Function
=================================
The Cumulative Distribution Function (CDF)
The Cumulative Distribution Function (CDF) of a random variable :math:`X` is a function :math:`F_X(x)` such that:
.. math::
F_X(x) &= P(X \leq x)
\lim_{x \to -\infty} F_X(x) &= 0
\lim_{x \to +\infty} F_X(x) &= 1
CDF can be used for:
.. math::
P(X \in ]a,b]) = P(a < X \leq b) = F_X(b) - F_X(a)
.. note::
In the definition above, the "less than or equal to" sign, ":math:`\le`", is a convention.
Since :math:`F_X(x)` is continuous on :math:`[0,1]` it is similar to :ref:`what is cover for the PDF <PDF>`.
It is not a universal convention but the distinction is important for discrete distributions.
More infos `here <https://en.wikipedia.org/wiki/Cumulative_distribution_function>`__.