0809CoreAReview - Info Defs Results Probability and Markov...

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Unformatted text preview: Info Defs Results Probability and Markov Chains – MATH 2561/2571 Lecturer Dr. O. Hryniv email [email protected] office CM309 http://www.dur.ac.uk/mathematical.sciences/teaching/... Info Defs Results This term we shall consider: • Review of Core A Probability • Generating functions • Markov chains • Elements of convergence and integration • . . . Info Defs Results Sample space & Events Sample space Ω is a collection of all possible outcomes of a probabilistic experiment; Event is a collection of possible outcomes, ie., a subset of the sample space. E.g., the impossible event ∅ , the certain event Ω; also, if A ⊂ Ω and B ⊂ Ω are events, one considers A ∪ B ( A or B ), A ∩ B ( A and B ), A c ≡ Ω \ A ( not A ), A \ B ( A but not B ). Info Defs Results σ-fields Definition Let A be a collection of subsets of Ω. We shall call A a field if it has the following properties: 1. ∅ ∈ A ; 2. if A 1 , A 2 ∈ A , then A 1 ∪ A 2 ∈ A ; 3. if A ∈ A , then A c ∈ A . Definition Let F be a collection of subsets of Ω. We shall call F a σ-field if it has the following properties: 1. ∅ ∈ F ; 2. if A 1 , A 2 , ··· ∈ F , then S ∞ k =1 A k ∈ F ; 3. if A ∈ F , then A c ∈ F . Info Defs Results Probability distribution Definition Let Ω be a sample space, and F be a σ-field of events in Ω. A probability distribution P on (Ω , F ) is a collection of numbers P( A ), A ∈ F , possessing the following properties: A1 for every event A ∈ F , P( A ) ≥ 0; A2 P(Ω) = 1; A3 for any pair of incompatible events A and B , P( A ∪ B ) = P( A ) + P(...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.

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0809CoreAReview - Info Defs Results Probability and Markov...

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