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Sirindhorn International Institute of TechnologyThammasat UniversitySchool of Information, Computer and Communication TechnologyEES315 2020/1Part IIDr.Prapun5Probability FoundationsConstructing the mathematical foundations of probability theoryhas proven to be a long-lasting process of trial and error.Theapproach consisting of defining probabilities as relative frequenciesin cases of repeatable experiments leads to an unsatisfactory theory.The frequency view of probability has a long history that goesback toAristotle. It was not until 1933 that the great Russianmathematician A. N.Kolmogorov (1903-1987) laid a satisfactorymathematical foundation of probability theory.He did this bytaking a number of axioms as his starting point, as had been donein other fields of mathematics. [21, p 223]We will try to avoid several technical details14 15in this class.Therefore, the definition given below is not the “complete” defini-tion. Some parts are modified or omitted to make the definitioneasier to understand.14To study formal definition of probability, we start with theprobability space,A, P).Let Ω be an arbitrary space or set of pointsω.Recall (from Definition 1.15) that, viewedprobabilistically, a subset of Ω is aneventand an elementωof Ω is asample point. Eachevent is a collection of outcomes which are elements of the sample space Ω.The theory of probability focuses on collections of events, called eventσ-algebras, typ-ically denoted byA(orF), that contain all the events of interest (regarding the randomexperimentE) to us, and are such that we have knowledge of their likelihood of occurrence.The probabilityPitself is defined as a number in the range [0,1] associated with each eventinA.15The class 2Ωof all subsets can be too large for us to define probability measures withconsistency, across all member of the class. (There is no problem when Ω is countable.)55
Definition 5.1. Kolmogorov’s Axioms for Probability:Aprobability measure16is a real-valued set function17that sat-isfiesP1Nonnegativity:P(A)0.P2Unit normalization:P(Ω) = 1.P3Countable additivityorσ-additivity: For every countablesequence (An)n=1of disjoint events,P[n=1An!=Xn=1P(An).The numberP(A) is called theprobabilityof the eventAFrom the three axioms18, we can derive many more propertiesof probability measure. These properties are useful for calculatingprobabilities.Definition 5.2.Some definitions involving events whose proba-bility = 1.

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