ECE 5510: Random Processes
Lecture Notes
Fall 2008
Lecture 2
Today: (1) Events as Sets (2) Axioms and Properties of Prob
ability (3) Experimental RSS Measurement
Announcements:
•
HW 1 assigned today, due a week from today.
•
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•
The application assignment 1 is posted on the wiki / WebCT.
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and then go to Teaching: ECE 5510. You can
also get there from WebCT.
1
Events as Sets
All probability is defined on sets. In probability, we call these sets
events
.
A set is a collection of elements.
In probability, we call
these
outcomes
.
Def’n:
Event
A collection of outcomes. Order doesn’t matter, and there are no
duplicates.
1.0.1
Set Terminology vs. Probability Terminology
Set Theory
Probability Theory
Probability Symbol
universe
sample space (certain event)
S
element
outcome (sample point)
s
set
event
E
disjoint sets
disjoint events
E
1
∩
E
2
=
∅
null set
null event
∅
1.1
Introduction
There are different ways to define an event (set):
•
List them:
A
=
{
0
,
5
,
10
,
15
,...
}
;
B
=
{
Tails,Heads
}
•
As an interval: [0
,
1], [0
,
1), (0
,
1), (
a,b
].
Note overlap with
coordinates!
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ECE 5510 Fall 2008
2
•
An existing event set name:
N
,
R
2
,
R
n
•
By rule:
C
=
{
x
∈
R
:
x
≥
0
}
,
D
=
{
(
x,y
)
∈
R
2
:
x
2
+
y
2
<
R
2
}
. Note Y&G uses ‘

’ instead of the colon ‘:’, which I find
confusing.
1.1.1
Important Events
Here’s an important event:
∅
=
{}
, the
null event
or the
empty set
.
Here’s the opposite:
S
is used to represent the set of everything
possible in a given context, the
sample space
.
•
S
=
B
above for the flip of a coin.
•
S
=
{
1
,
2
,
3
,
4
,
5
,
6
}
for the roll of a (6sided) die.
•
S
=
{
Adenine,Cytosine,Guanine,Thymine
}
for the nucleotide
found at a particular place in a strand of DNA.
•
S
=
C
,
i.e.
nonnegative real numbers, for your driving speed
(maybe when the cop pulls you over).
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 Fall '08
 Chen,R
 Probability theory

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