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Unformatted text preview: ORIE 3500/5500 Fall Term 2008 Review 1 1 Probability Model In any experiment we first consider the probability space. 1.1 Probability Space It consists of three things: • Sample space Ω, which is the set of all possible outcomes, • Collection of events F , • Probability P : F → [0 , 1]. P assigns values to each event and should satisfy three basic axioms: 1. Nonnegativity: P ( A ) ≥ 0 for any event A . 2. Additivity: P ( A 1 ∪ A 2 ∪ ... ) = P ( A 1 ) + P ( A 2 ) + ··· for disjoint sets A 1 ,A 2 ,... . 3. Normalization: P (Ω) = 1. Using these axioms one can easily obtain 1. P ( A c ) = 1 P ( A ) for any event A . 2. P ( A ∪ B ) = P ( A ) + P ( B ) P ( A ∩ B ) for any events A and B . 3. If Ω is finite and all the outcomes are equally likely, then P ( A ) =  A  /  Ω  for any event A . 1.2 Conditional Probability and Independence Suppose P ( B ) > 0 and the event B has occurred. We want to use this information to find the probability of A . The conditional probability of A given B is defined as P ( A  B ) = P ( A ∩ B ) P ( B ) = Q B ( A ) , (say) . Note that Q B ( · ) is itself a probability law, i.e. it satisfies all the axioms of probability. Moreover Q B ( B ) = 1 and Q B ( C ) = 0 for any C ⊂ B c . 1 So Q B ( · ) represents an universe where the event B occurs for sure. Note that the effect of repeated conditioning is Q B ( A  D ) = Q B ( A ∩ D ) Q B ( D ) = P ( A  B ∩ D ) = Q BD ( A ) . Two events A and B are independent if Q B ( A ) = P ( A ) ⇔ P ( A ∩ B ) = P ( A ) P ( B ) , i.e. the happening of B has no effect on the happening of A . Using analogy a collection of events A 1 ,...,A n will be independent if for any subset S ⊂ { 1 , 2 ,...,n } , of indices P ( ∩ i ∈ S A i ) = Y i ∈ S P ( A i ) . An important and useful consequence is as follows. If A 1 ,...,A n are in dependent and if S and T are disjoint subsets of indices, i.e. S,T ⊂ { 1 , 2 ,...,n } and S ∩ T = ∅ , then any events B and C involving oper ations (such as union, intersection or complement) of { A i : i ∈ S } and { A i : i ∈ T...
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 Fall '08
 WEBER
 Probability theory, CDF

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