We now define the concept of a random variable, a key definition in
probability and statistics and for statistical simulation in general.
We define a random
variable as a real number 
that is assigned to an event 
.
It is random
because the event is random, and it is variable because the assignment of
the value may vary over the real axis.
We will use ``r.v.'' as an abbreviation for
``random variable''.
Assign the number 
to each face 
of a die. When face
appears, the r.v. is .
Random variables are useful because they allow the quantification of random
processes, and they facilitate numerical manipulations, such as the definition
of mean and standard deviation, to be introduced below.
For example, if one
were drawing balls of different colors from a bowl, it would be difficult to
envision an ``average'' color, although if numbers were assigned to the
different colored balls, then an average could be computed.
On the other hand, in
many cases of real interest, there is no reasonable way to assign a real
number to the outcome of the random process, such as the outcome of the
interaction between a 1 eV neutron and a uranium-235 nucleus, which might lead
to fission, capture, or scatter.
In this case, defining an ``average'' interaction
makes no sense, and assigning a real number to the random process does not
assist us in that regard.
Nevertheless, in the following discussion, we have tacitly assumed a real number has been assigned to the event that we know occurs with probability 
.
Thus, one can in essence say that the r.v. occurs with probability .