=================
 Random Variables
=================

Remember the random experiment of throwing with a die? We defined
events then as "6" or "odd". The events "1",...,"6" of course are the
elementary events in the universe $U$ of all possible outcomes. Note
that we write "1" here instead of 1 to stress that we are labeling
events.

Let $u\in U$ be such an elementary outcome in the universe $U$. Then
we can set up a mapping $X: u\in U\mapsto X(u)\in B$ where $B$ is
either $\setZ$ or $\setR$. Such a mapping is called a **random
variable**. A RV is a numerical observation of the outcome of a random
experiment.

Consider again the experiment of throwing with a fair die. In this
case $u\in\{\text{"1"}, \text{"2"}, \ldots,\text{"6"}\}$. An obvious
mapping in this case is $X(\text{"1"}) = 1$, $X(\text{"2"}) = 2$
etc. This is an example of a **discrete random variable**, the range
of the mapping $X$ is the set of integers $\setZ$.

Now consider the experiment where we measure the length of a randomly
selected person, so $U$ is the set of all persons. The length can be
any real number in the range of $0$ to $\infty$ (of course not all
lengths are equally probable). The random variable in this case is
$X(u) = \text{length of person } u$, i.e. $X: u\in U \mapsto
\setR$. Such a RV is called a **continuous random variable**.


.. toctree::
   :hidden:

   rvDiscrete
   rvContinuous
   rvCDF
   rvExpVar
   rvDistributions
   rvConditional