This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 σfield 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 σfield in Ω is just braceleftbig ∅ , Ω bracerightbig and the biggest σfield consists of all subsets of Ω. We observe the following simple fact: Exercise 0.3 Show that if F 1 and F 2 are σfields, then 1 F 1 ∩ F 2 is a σfield, but, in general, F 1 ∪ F 2 is not a σfield. 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 σfields; i O.H. Probability and Markov Chains – MATH 2561/2571 E09 Definition 0.4 Let Ω be a sample space, and F be a σfield 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 nontrivial only in examples with infinitely many different events, ie., when the collection F of all events (and, therefore, the sample space Ω ) is infinite....
View
Full
Document
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.
 Spring '10
 Various

Click to edit the document details