# Introduction; Review of Random Variables

Probability :: Set Theory :: Definition :: Conditional :: Random :: Distribution :: R.V.S.

## Random variables

Up to this point, the outcomes in
could be anything: they
could be elephants, computers, or mitochondria, since
is
simple expressed in terms of sets. But we frequently deal with
*
numbers
*
, and want to describe events associated with sets of
numbers. This leads to the idea of a random variable.

A function
such that
, that is, such that the events involved
are in
, is said to be
**
measurable
**
with respect to
.
That is,
is divided into sufficiently small pieces that the
events in it can describe all of the sets associated with
.

We observe that a random variable cannot generate partitions of the
underlying sample space which are not events in the
-field
.

Another way of saying that
is measurable: For any
,

Recalling the idea of Borel sets
associated with the real line,
we see that a random variable is a measurable function from
to
:

We will abbreviate the term random variable as

**r.v.**.

We will use a notational shorthand for random variables.

By this definition, we can identify a new probability space. Using
as the pre-probability space, we use the measure

So we get the probability space .

As a matter of practicality, if the sample space is
, with the
Borel field,
*
most
*
mappings to
will be random variables.

To summarize:

where

for .