notes-ch3-p49-p68-annotated.pdf - Chapter 3 Discrete Random Variables and Probability Distributions 3.1 Random Variables Random variable(rv \u2013 is

notes-ch3-p49-p68-annotated.pdf - Chapter 3 Discrete Random...

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Chapter 3 Discrete Random Variables and Probability Distributions 3.1 Random Variables Random variable (rv) – is obtained by assigning a numerical value to each outcome in the sample space S of a particular experiment. In mathematical language, a random variable is a function whose domain is the sample space and whose range is the set of real numbers. Random variables are denoted by uppercase letters such as X and Y . A discrete random variable is a random variable that takes either a finite number of possible values or at most a count- ably infinite number of possible values. A continuous random variable is a random variable that takes an infinite number of possible values that is not count- able. For any possible value c , P ( X = c ) = 0 . 49
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Chapter 3. Discrete Random Variables and Probability Distributions STAT 155 In this chapter, we examine the basic properties and discuss the most important examples of discrete variables. Chapter 4 focuses 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
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Chapter 3. Discrete Random Variables and Probability Distributions STAT 155 3.2 Probability Distributions for Discrete Random Variables Probability distribution – A table, graph, or mathematical for- mula that provides the possible values of a random variable X and their corresponding probabilities. The probability distribution or probability mass function (pmf) of a discrete variable X is defined for every number x by p ( x ) = P ( X = x ) = P ( all s 2 S ; X ( s ) = x ) , which is the sum of the probabilities of all sample points in S that are assigned the value x . 0 p ( x ) 1 P x p ( x ) = 1 The cumulative distribution function (cdf) F ( x ) of a discrete random variable X is defined for every number x by F ( x ) = P ( X x ) = P y : y x p ( y ) . 0 F ( x ) 1 F ( a ) = 0 , for a < x min , x min is the smallest possible X value F ( b ) = 1 , for b x max , x max is the largest possible X value 51
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Chapter 3. Discrete Random Variables and Probability Distributions STAT 155
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