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

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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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