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
}
.
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- Spring '10
- Lehman
- Probability theory, Ω, Countable set, countable number, H1 T2
-
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