Introduction; Review of Random Variables
Probability :: Set Theory :: Definition :: Conditional :: Random :: Distribution :: R.V.S.
Definition of probability
We now formally define what we mean by a probability space. A probability space has three components.
The first is the
sample space
, which is the collection of all
possible outcomes of some experiment. The outcome space is frequently
denoted by
.
We deal with
subsets
of
. For example, we might have an
event which is "all even throws of the die'' or "all outcomes
''. We denote the collection of subsets of interest as
. The
elements of
(that is, the subsets of
) are called
events
.
is called the
event class
.
We will
restrict
to be a
field
.
The tuple is called a preprobability space (because we haven't assigned probabilities yet). This brings us to the third element of a probability space: Given a preprobability space , a probability distribution or a measure on is a mapping from to (which we will write: ) with the properties:
 (this is a normalization that is always applied for probabilities, but there are other measures which don't use this)
 . (Measures are nonnegative)

If
such that
for
all
(that is, the sets are "disjoint'' or "mutually
exclusive'') then
The triple is called a probability space .
 tells what individual outcomes are possible
 tells what sets of outcomes  events  are possible.
 tells what the probabilities of these events are.
Some properties of probabilities (which follow from the axioms):

.


.


If
then
We write this as ( converges up to ). Similarly, if
(a decreasing sequence), define
If , then . If , then .
We will introduce more properties later.
Some examples of probability spaces

(a discrete set of outcomes).
. Let
be a set of
nonnegative numbers satisfying
. Define the
function
by

Uniform distribution: Let
,
=smallest
field containing all intervals in [0,1]. We can
take (without proving that this actually works  but it does)