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

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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/...

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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 ).

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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( B ); A4 for any countable collection A 1 , A 2 , . . . of mutually

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

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