Discrete Random Variables
=========================


A discrete random variable $X$ maps the elementary events $u$ in $U$
unto a value in $\setZ$. i.e. $X: u\in U \mapsto X(u)\in\setZ$. Note
that the random variable $X$ in iteself is not an event. We can
specify an event $A$ by specifying

.. math::
   A = \{ u \bigm| X(u)=x \}

We will most often abbreviate this and write the event $A$ as
$X=x$. Carefully note the distinction between capital $X$ and $x$. The
capital $X$ always refers to random variables and lower case letters
refer to values from the range of the mapping $X$.

A discrete variable is completely characterized in case for all
$x\in\setZ$ we specify $\P(X=x)$. We will do this with the
**probability mass functions** $p_X$:

.. math::
   p_X(x) = \P(X=x)

As an example consider our fair dice again, there we have:

.. math::
   p_X(x) = \begin{cases}
   0 &: x < 1\\
   \frac{1}{6} &: 1\leq x \leq 6\\
   0 &: x > 6
   \end{cases}


.. ipython:: python
   :suppress:
      
   x = np.arange(-5, 12)
   p = np.zeros_like(x, dtype=float)
   p[logical_and(x>=1, x<=6)] = 1/6
   plt.stem(x,p)
   @savefig pmsdice.png
   plt.show()


Note that we assume that the events $X=x_1$ and $X=x_2$ are disjunct
in case $x_1\not=x_2$ and futhermore that all $X=x$ for $x\in\setZ$
form a partition of $U$ and therefore:

.. math::
   \sum_{x\in\setZ} p_X(x) = 1