ECE 6010
Lecture 1 – Introduction; Review of Random Variables
: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1,
Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section 4.4, Section 4.5
Why study probability?
1. Communication systems. Noise. Information.
2. Control systems. Noise in observations. Noise in interference.
3. Computer systems. Random loads. Networks. Random packet arrival times.
Probability can become a powerful engineering tool. One way of viewing it is as “quantified
common sense.” Great success will come to those whose tool is sharp!
Set theory
Probability is intrinsically tied to set theory. We will review some set theory concepts.
We will use
c
to denote complementation of a set with respect to its universe.
– union:
A
B
is the set of elements that are in
A
or
B
.
– intersection:
A
B
is the set of elements that are in
A
and
B
. We will also denote
this as
AB
.
a
A
:
a
is an element of the set
A
.
A
B
:
A
is a subset of
B
.
A
B
:
A
B
and
B
A
.
Note that
A
A
c
(where
is the universe).
Notation for some special sets:
R
– set of all real numbers
Z
– set of all integers
Z
– set of all positive integers
N
– set of all natural numbers, 0,1,2,…,
R
n
– set of all
n
tuples of real numbers
C
– set of complex numbers
Definition 1
A
field
(or algebra) of sets is a collection of sets that is closed under comple
mentation and finite union.
2
That is, if
F
is a field and
A
F
, then
A
c
must also be in
F
(closed under complemen
tation). If
A
and
B
are in
F
(which we will write as
A
,
B
F
) then
A
B
F
.
Note: the properties of a field imply that
F
is also closed under finite intersection.
(DeMorgan’s law:
AB
(
A
c
B
c
)
c
)
Definition 2
A
σ
field (or
σ
algebra) of sets is a field that is also closed under
countable
unions (and intersections).
2
What do we mean by countable?
• A set with a finite number in it is countable.
• A set whose elements can be matched oneforone with
Z
is countable (even if it has
an infinite number of elements!)
Are there noncountable sets?
Note: For any collection
F
of sets, there is a
σ
field containing
F
, denoted by
σ (
F
)
.
This is called the
σ
field generated by
F
.
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2
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
.
Example 1
Suppose the experiment involves throwing a die.
1
,
2
,
3
,
4
,
5
,
6
2
We deal with
subsets
of
. For example, we might have an event which is “all even throws
of the die” or “all outcomes
4”. We denote the collection of subsets of interest as
F
. The
elements of
F
(that is, the subsets of
) are called
events
.
F
is called the
event class
.
We
will restrict
F
to be a
σ
field
.
Example2
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 Spring '08
 Stites,M
 Probability, Probability theory, CDF

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