CHAPTER 3 Probabilities Associated with Random Variables
We will revisit random variables again in the next chapter. But now it is time to complete
the overall picture. We started with the notion of a random variable and its associated
sample space. We then brought in the concept of an event, which is, simply, a subset of
the sample space. Now we will complete the picture with an overview of probability.
Once one has identified an event of interest, it is natural to ask: what is the probability
that this event will occur? And so, in a way, the probability of an event is a kind of
measure of the
size
of the event. But it is not size in the typically interpreted manner. We
will begin this chapter with a discussion of what is meant by the term
size
, as related to
the probability of an event. We will then proceed to more formally develop the notion of
a probability measure associated with a random variable.
.
3.1 Probability as a Measure of Size
When most people think of the size of an object, they think of physical size. Size is an
indicator of how small or large an object is. When the object is a subset of
n
-dimensional
Euclidean space,
n
ℜ
, then size is easily quantified; that is, it can be assigned a number.
For example, consider an
n
-dimensional cube whose side has length, λ. It’s Euclidean
size is then
n
λ
. For
n=
1, this is simply a line of length λ. For
n
=2 it is a square with area
2
. In 3-dimensional space it is a cube with volume
3
. For
n
>3 it can no longer be
visualized, but we can extend the notion of size, and call it a hyper cube with “volume”
n
. Okay. So, what are some fundamental properties of this entity called
size
? Well, we
typically require size to be non-negative. Also, if a set is contained in another set, then
the size of the former can be no bigger than the size of the latter. Still another reasonable
property to require is that the size of two or more nonintersecting sets must equal the sum
of the sizes of the individual sets. Now we are in a position to list the properties that the
measure of size called probability must have.
Definition 3.1
A measure of the size of events contained in the field of events,
X
Ω
,
associated with a random variable,
X
, call it Pr(▪), is a
probability measure
if it satisfies
all of the following properties: