EECS 501
PROBABILITY MAPPING
Fall 2001
DEF:
Ω=
sample space
=set of all distinguishable outcomes of an experiment.
DEF:
A
=
event space
=set of subsets of Ω such that
A
is a
σ

algebra
.
DEF:
An
Algebra
=
A
=a set of subsets of a set Ω such that:
1.
A
∈ A
and
B
∈ A →
A
∪
B
∈ A
and
A
∩
B
∈ A
;
2.
A
∈ A →
A
0
= Ω

A
∈ A
. Closed under
∪
,
∩
, complement in Ω.
DEF:
A
σ

algebra
is an algebra closed under
countable
number of
∪
,
∩
.
NOTE:
Empty set=
φ
and Ω are always members of any algebra
A
,
since
A
∈ A →
A
0
∈ A →
φ
=
A
∩
A
0
∈ A
and Ω =
A
∪
A
0
∈ A
.
NOTE:
A
∈ A
and
B
∈ A →
A
∪
B
∈ A
and
A
∩
B
= (
A
0
∪
B
0
)
0
∈ A
.
So DeMorgan’s law
→
closure under
∪
and
0
→
closure under
∩
.
EX: Experiment:
Flip a coin twice. Let
H
i
=heads on
i
th
flip.
Sample space:
Ω =
{
H
1
H
2
, H
1
T
2
, T
1
H
2
, T
1
T
2
}
(2
2
elements).
Event space:
A
=power set of Ω=set of all subsets of Ω (2
2
2
elements).
A
=
{
φ,
Ω
,
{
H
1
H
2
}
,
{
H
1
T
2
}
,
{
T
1
H
2
}
,
{
T
1
, T
2
}
,
{
H
1
}
,
{
H
2
}
,
{
T
1
}
,
{
T
2
}
,
{
H
1
H
2
}∪{
T
1
T
2
}
,
{
H
1
T
2
}∪{
H
2
T
1
}
,
{
H
1
H
2
}
0
,
{
H
1
T
2
}
0
,
{
T
1
H
2
}
0
,
{
T
1
T
2
}
0
}
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Lehman
 Probability theory, Ω, Countable set, countable number, H1 T2

Click to edit the document details