0809CoreAProbability2H - O.H. Probability and Markov Chains...

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Unformatted text preview: O.H. Probability and Markov Chains – MATH 2561/2571 E09 Preliminaries This section revises some parts of Core A Probability, which are essential for this course, and lists some other mathematical facts to be used (without proof) in the following. Probability space We recall that a sample space Ω is a collection of all possible outcomes of a probabilistic experiment; an event is a collection of possible outcomes, ie., a subset of the sample space. We introduce the impossible event ∅ and the certain event Ω; also, if A ⊂ Ω and B ⊂ Ω are events, it is natural to consider other events such that A ∪ B ( A or B ), A ∩ B ( A and B ), A c ≡ Ω \ A ( not A ), and A \ B ( A but not B ). Definition 0.1 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 . Remark 0.1.1 Obviously, every field is closed w.r.t. taking finite unions or intersections. Definition 0.2 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 uniontext ∞ k =1 A k ∈ F ; 3. if A ∈ F , then A c ∈ F . Remark 0.2.1 Obviously, property 2 above can be replaced by the equivalent condition intersectiontext ∞ k =1 A k ∈ F . Clearly, if Ω is fixed, the smallest σ-field in Ω is just braceleftbig ∅ , Ω bracerightbig and the biggest σ-field consists of all subsets of Ω. We observe the following simple fact: Exercise 0.3 Show that if F 1 and F 2 are σ-fields, then 1 F 1 ∩ F 2 is a σ-field, but, in general, F 1 ∪ F 2 is not a σ-field. If A and B are events, we say that A and B are incompatible (or disjoint), if A ∩ B = ∅ . 1 and, in fact, an intersection of arbitrary (even uncountable !) collection of σ-fields; i O.H. Probability and Markov Chains – MATH 2561/2571 E09 Definition 0.4 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 ) ≥ ; 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 incompatible 2 events, P parenleftBig ∞ uniondisplay k =1 A k parenrightBig = ∞ summationdisplay k =1 P ( A k ) . Remark 0.4.1 Notice that the additivity axiom A4 above does not extend to uncountable collections of incompatible events. Remark 0.4.2 Obviously, property A4 above and Definition 0.2 are non-trivial only in examples with infinitely many different events, ie., when the collection F of all events (and, therefore, the sample space Ω ) is infinite....
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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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0809CoreAProbability2H - O.H. Probability and Markov Chains...

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