01prob - EECS 501 PROBABILITY MAPPING Fall 2001 DEF =sample...

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
} .