Minor changes

source/statistics/probability_distribution_functions.rst

@@ -1,4 +1,46 @@
 Probability Distribution Functions
 ------------------------------------
 
-todo CDF and PDF
+Probability Density Function
+=============================
+
+The Probability Density Function (PDF) is function defined for a random variable :math:`X` such that:
+
+.. math::
+   \forall (a,b) \in \mathbb{R}^2,\, P(a \le X \le b) = \int_a^b f_X(x)dx
+
+Properties:
+
+#. :math:`\int_{-\infty}^{+\infty} f_X(x)dx=1`
+#. :math:`P(X=a)=\int_{a}^{a} f_X(x)dx=0`
+
+From property *2* it can be derived that (`source <http://yallouz.arie.free.fr/terminale_cours/probascont/prob-continue.php>`__):
+
+.. math::
+   P(a \le X \le b) &= P(a < X \le b) =P(a \le X < b) =P(a < X < b)
+
+   P(a \ge X) &= P(a > X) = 1-P(a < X) = 1-P(a \le X)
+
+The PDF of a random variable is intimately related to its :ref:`CDF <CDF>` with the following relation:
+
+.. math::
+   F_X(x)=\int_{-\infty}^x f_X(t)dt
+
+To illustrate this property let's take an example with the exponential distribution defined as follow:
+
+.. math::
+    f(x;\lambda) = \begin{cases}\lambda  e^{ - \lambda x} & x \ge 0,\\ 0 & x < 0.\end{cases}
+    
+Let's compute its CDF:
+
+.. math::
+   F(x;\lambda)=\int_{0}^x \lambda e^{-\lambda t}dt = -\int_{0}^x -\lambda e^{-\lambda t}dt &= - \left[ e^{-\lambda t} \right]_{0}^x
+
+   &=1 -e^{-\lambda x}
+   
+.. _CDF:
+
+Cumulative Distribution Function
+=================================
+
+The Cumulative Distribution Function (CDF)