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Section 3 - Notes Stats 160.pdf - STATISTICS 160 - Fall...

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STATISTICS 160 - Fall 2022J.H. McVittieSection 3 - Course NotesThe Random VariableIn many settings, sample points are associated to real number values.Byextending our sample point methodology to the real line, we will have themathematical tools needed to compute the probabilities of more complicatedevents and generalize these events to a wide variety of applications.Definition.Arandom variableXis a function which maps values fromthe sample spaceSto the set of all real numbersR:X:SRWe denoteRXas the range ofX(i.e. the set of all values thatXcan possiblymap to).Definition.A random variable is said to bediscreteif the rangeRXis madeup of either a finite or countable infinite number of distinct values.For discrete random variables, since the range is finite or countably infinite,then we can characterize the probability that a discrete random variableXtakes on a particular valuexthrough the expressionP(X=x) for allxRX.We formalize this in the following definition:Definition.Theprobability distributionfor a discrete random variableYcan be represented by a formula which providesp(y) =P(Y=y)for ally.It is a convention that any valueynot explicitly assigned a positive prob-ability is understood to be zero (i.e.p(y) = 0). Essentially, the probabilitydistribution of a random variableYindicates how the total probability 1 is“distributed” among all the possible values that the random variableYmapsto.1
STATISTICS 160 - Fall 2022J.H. McVittieExample:LetYbe the sum obtained by rolling a pair of die. By utilizingthe tools from the first two lectures, one can form the following table bycounting the number of ways in which each value ofyis obtained and dividingby the size of the sampleS:y23456789101112P(Y=y)1/362/363/364/365/366/365/364/363/362/361/36In terms of the random variableY, the above table corresponds to the functionP(Y=y) = (6- |y-7|)/36 fory∈ {2,3, ...,12}.We give a name to thefunctionP(Y=y) in the following section.The Probability Mass Function, Expected Value, Variance and Stan-dard DeviationDefinition.For a discrete random variableY, the function which providesp(y) =P(Y=y)for ally, is defined as theprobability mass functionorpmf. According to the Kolmogorov Axioms, the functionp(y)must satisfy:p(y)0for allyyp(y) = 1where the summation is over all values ofyA particular quality of a distribution which is usually of interest to re-searchers is a measure of “center” or the “balancing point” of a distribution.

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Probability theory, discrete random variable Y, J H McVittie

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