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Chapter 3Discrete Random Variables andProbability Distributions3.1 Random VariablesRandom variable (rv)– is obtained by assigning a numericalvalue to each outcome in the sample spaceSof a particularexperiment. In mathematical language, a random variable is afunction whose domain is the sample space and whose rangeis the set of real numbers. Random variables are denoted byuppercase letters such asXandY.•Adiscrete random variableis a random variable that takeseither a finite number of possible values or at most a count-ably infinite number of possible values.•Acontinuous random variableis a random variable thattakes an infinite number of possible values that is not count-able. For any possible valuec,P(X=c) = 0.49
Chapter 3. Discrete Random Variables and Probability DistributionsSTAT 155In this chapter, we examine the basic properties and discussthe most important examples of discrete variables. Chapter 4focuses on continuous random variables.Exercise 3.7For each random variable defined here, describe the set ofpossible values for the variable, and state whether the variable is discrete.(a)X= the number of unbroken eggs in a randomly chosen standard eggcarton(b)Y= the number of students on a class list for a particular course whoare absent on the first day of classes(c)U= the number of times a duffer has to swing at a golf ball beforehitting it(d)X= the length of a randomly selected rattlesnake(e)Z= the amount of royalties earned from the sale of a first edition of10,000 textbooks(f)Y= the pH of a randomly chosen soil sample(g)X= the tension (psi) at which a randomly selected tennis racket hasbeen strung(h)X= the total number of coin tosses required for three individuals toobtain a match (HHH or TTT)50
Chapter 3. Discrete Random Variables and Probability DistributionsSTAT 1553.2 Probability Distributions for Discrete RandomVariablesProbability distribution– A table, graph, or mathematical for-mula that provides the possible values of a random variableXand their corresponding probabilities.Theprobability distributionorprobability mass function (pmf)of a discrete variableXis defined for every numberxbyp(x) =P(X=x) =P(alls2S;X(s) =x),which is the sum of the probabilities of all sample points inSthat are assigned the valuex.•0p(x)1•Pxp(x) = 1Thecumulative distribution function (cdf)F(x)of a discreterandom variableXis defined for every numberxbyF(x) =P(Xx) =Py:yxp(y).•0F(x)1•F(a) = 0, fora < xmin,xminis the smallest possibleXvalue•F(b) = 1, forb≥xmax,xmaxis the largest possibleXvalue51
Chapter 3. Discrete Random Variables and Probability DistributionsSTAT 155