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chap03 - Chapter 3 Axiomatic approach to probability...

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Chapter 3 Axiomatic approach to probability Chapter Overview: Axioms of probability and terminolgy Basic probability theorems Special cases of probability space: - Discrete (finite and countably infinite) - Continuous (uncountably infinite) 48
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3.1 Axioms of probability 49 3.1 Axioms of probability Random experiment: An experiment, either natural or man-made, in which one among several identified results are possible, is called a random experiment . The possible results of the experiments are called outcomes . A particular realization of the experiment, leading to a particular out- come, is called a trial . Probability space: In the axiomatic approach to probability, a random experiment is mod- eled as a probability space , the latter being a triplet ( S, F , P ), where - S is the sample space, - F is the set of events (events algebra), - P ( . ) is the probability function. These concepts are described individually below. c 2003 Benoˆ ıt Champagne Compiled February 2, 2012
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3.1 Axioms of probability 50 Sample space: The sample space S is the set of all possible results, or outcomes, of the random experiment. In practical applications, S is defined by the very nature of the problem under consideration. S may be finite, countably infinite or uncountably infinite. The elements of S , i.e. the experimental outcomes, will usually be de- noted by lower case letters (e.g.: s, a, x, etc...) Example 3.1: I Consider a random experiment that consists in flipping a coin twice. A suitable sample space may be defined as S = { HH, HT, TH, TT } where, for example, outcome HT corresponds to heads on the first toss and tails on the second. Here, S is finite with only 4 outcomes. J c 2003 Benoˆ ıt Champagne Compiled February 2, 2012
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3.1 Axioms of probability 51 Events: In probability theory, an event A is defined as a subset of S , i.e. A S . Referring to a particular trial of the random experiment, we say that A occurs if the experimental outcome s A . Special events S and : - Since for any outcome s , we have s S by definition, S always occurs and is thus called the certain event. - Since for any outcome s , we have s 6∈ ∅ , never occurs and is thus called the impossible event. Example 3.1 (continued): I Consider the event A = { getting heads on the first flip } . This can equivalently be represented by the following subset of S : A = { HH, HT } ⊂ S Let s denote the outcome of a particular trial: if s = HH or HT A occurs if s = TH or TT A does not occur J c 2003 Benoˆ ıt Champagne Compiled February 2, 2012
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3.1 Axioms of probability 52 Events algebra: Let F denote the set of all events under consideration in a given random experiment. Note that F is a set of subsets of S Clearly: - F must be large enough to contain all interesting events, - but not so large as to contain impractical events that lead to math- ematical difficulties. (This may be the case when S is uncountably infinite, e.g. S = R n .) In the axiomatic approach to probability, it is required that F be a σ -algebra: (a) S ∈ F (b) A ∈ F ⇒ A c ∈ F (c) A 1 , A 2 , ... ∈ F ⇒ ∪ i A i ∈ F
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