Notes on Probability Theory and Statistics

Notes on Probability Theory andStatisticsAntonis Demos(Athens University of Economics and Business)October 20022 Part IProbability Theory3 Chapter Set Theory DigressionAsetis defined as any collection of objects, which are biggest possible collection of points under consideration is called thespace,universe,oruniversal set. For Probability Theory the space is called called asubsetofB(we writeA BorB A) if every elementofAis also an element called aproper subsetofB(we writeA BorB A) if every element ofAis also an element ofBand there is at least one elementofBwhich does not belong setsAandBare calledequivalent setsorequal sets(we writeA=B)ifA BandB a set has no points, it will be called theemptyornullset and denoted by .Thecomplementof a setAwith respect to the space , denoted by A,Ac,or A, is the set of all points that are in but not two setsAandBis a set that consists of the commonelements of the two sets and it is denoted byA two setsAandBis a set that consists of all points that are inAorBor both (but only once) and it is denoted byA differenceof two setsAandBis a set that consists of all points in6 IntroductionAthat are not inBand it is denoted byA of Set

Frequency or a posteriori Probability : Is the ratio of the number αthat an event Ahas occurred out of ntrials, i.e. P(A)=α/n. Example: Assume that we flip a coin 1000 times and we observe 450 heads. Then the a posteriori probability is P(A)=α/n=450/1000 …

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