Random variables are introduced for convenient description of experiments with
numerical outcomes. (The other option is to select 
, or
.) If we want to run computer simulations, we need to
represent even non-numerical experiments (like tossing coins) in numerical terms
anyhow. Thus the language of random variables becomes the natural extension of
elementary probability theory, expressing many of the same concepts in a little
different language.
A random variable is the numerical quantity assigned to every outcome of the
experiment. In mathematical terms, random variable is a function 
with the property that sets 
are events in
for all 
. Recall that the last conditions means that we
may talk about probabilities of events .
Probabilities for a one-dimensional r. v. are determined by the cumulative distribution function
The corresponding tail function 
is sometimes called the
reliability
function .
Cumulative distribution function can be used to express probabilities of intervals
. Since probability is continuous, (
) we can also
compute 
. The right hand side limit
exists, as F is a non-decreasing function.
In probability theory we are concerned with probabilities. Random variables that
have the same probabilities are therefore considered equivalent. We write 
to denote the equality of distributions , ie.
for all Borel sets 
(say, all
intervals U).
Vector valued r. v. are measurable the functions 
. In
the vector case we also refer to 
as the d-variate, or
multivariate, random variable.
We will use the ordinary notation for sums and inequalities between random
variables. There is however a word of caution. In probability theory, equalities
and inequalities between random variables are interpreted almost surely. For
instance 
means 
; the latter is a shortcut that we
use for the expression 
.
