Chapter 03

# Chapter 03 - Chapter 3 Axiomatic Approach to Probability...

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Chapter 3: Axiomatic Approach to Probability 3.1 Axioms of Probability B. Champagne 1 1 Department of Electrical & Computer Engineering McGill University September 4, 2009 ECSE 305 3.1 Axioms of Probability

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Outline 3.1 Axioms of probability and terminology 3.2 Basic probability theorems 3.3 Special cases of probability space: Discrete (ﬁnite and countably inﬁnite) Continuous (uncountably inﬁnite) ECSE 305 3.1 Axioms of Probability
3.1 Axioms of Probability Random experiment: An experiment 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 outcome, is called a trial . Probability space: In the axiomatic approach to probability, a random experiment is modeled as a probability space , i.e. a triplet ( S , F , P ) , where - S is the sample space, - F is the set of events (events algebra), - P ( . ) is the probability function. ECSE 305 3.1 Axioms of Probability

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Sample Space The sample space S is the set of all possible results, or outcomes, of the random experiment. The choice of S is dictated by the nature of the problem. S may be ﬁnite, countably inﬁnite or uncountably inﬁnite. The elements of S , i.e. the experimental outcomes, will usually be denoted by lower case letters (e.g.: s , a , x , etc. ..) Example 3.1 Consider a random experiment that consists in ﬂipping a coin twice. A suitable sample space may be deﬁned as S = { HH , HT , TH , TT } For instance, outcome HT corresponds to heads on the ﬁrst toss and tails on the second. Here, S is ﬁnite with only 4 outcomes. ECSE 305 3.1 Axioms of Probability
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 , 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. ECSE 305 3.1 Axioms of Probability

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Example 3.1 (cont.) Consider the random experiment consisting in ﬂipping a coin twice, for which a suitable sample space is S = { HH , HT , TH , TT } Consider the event A = { getting heads on the ﬁrst ﬂip } . This can equivalently be represented by the subset: A = { HH , HT } ⊂ S Let s denote the outcome of a particular trial: if s = HH or HT , we say that A occurs; if s = TH or TT , then A does not occur. ECSE 305 3.1 Axioms of Probability
Event Algebra: Let F denote the set of all events under consideration in a given random experiment. F must be large enough to contain all interesting events; but not so large as to contain impractical events that lead to mathematical difﬁculties. In the axiomatic approach to probability, F is required to be a σ -algebra: (a) S ∈ F (b) A ∈ F ⇒ A c ∈ F (c) A 1 , A 2 ,...

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Chapter 03 - Chapter 3 Axiomatic Approach to Probability...

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