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Unformatted text preview: 2 Random Variables 2.1 Introduction 59 2.2 Definition of a Random Variable 59 2.3 Events Defined by Random Variables 61 2.4 Distribution Functions 62 2.5 Discrete Random Variables 63 2.6 Continuous Random Variables 70 2.7 Chapter Summary 75 2.8 Problems 76 2.1 Introduction The concept of a probability space that completely describes the outcome of a random experiment has been developed in Chapter 1. In this chapter we develop the idea of a function defined on the outcome of a random experiment, which is a very highlevel definition of a random variable. Thus, the value of a random vari able is a random phenomenon and is a numerically valued random phenomenon. 2.2 Definition of a Random Variable Consider a random experiment with sample space S . Let w be a sample point in S . We are interested in assigning a real number to each w S . A random variable, X ( w ) , is a singlevalued real function that assigns a real number, called the value of X ( w ) , to each sample point w S . That is, it is a mapping of the sample space onto the real line. 59 60 Chapter 2 Random Variables Generally a random variable is represented by a single letter X instead of the function X ( w ) . Therefore, in the remainder of the book we use X to denote a ran dom variable. The sample space S is called the domain of the random variable X . Also, the collection of all numbers that are values of X is called the range of the random variable X . Figure 2.1 illustrates the concepts of domain and range of X . Example 2.1 A cointossing experiment has two sample points: heads and tails. Thus, we may define the random variable X associated with the experiment as follows: X ( heads ) = 1 X ( tails ) = In this case, the mapping of the sample space to the real line is as shown in Fig ure 2.2. trianglesolid Figure 2.1 A Random Variable Associated with a Sample Point Figure 2.2 Random Variable Associated with CoinTossing Experiment 2.3 Events Defined by Random Variables 61 2.3 Events Defined by Random Variables Let X be a random variable and x be a fixed real value. Let the event A x define the subset of S that consists of all real sample points to which the random variable X assigns the number x . That is, A x = { w  X ( w ) = x } = [ X = x ] Since A x is an event, it will have a probability, which we define as follows: p = P [ A x ] We can define other types of events in terms of a random variable. For fixed numbers x , a , and b , we can define the following: [ X x ] = { w  X ( w ) x } [ X > x ] = { w  X ( w ) > x } [ a < X < b ] = { w  a < X ( w ) < b } These events have probabilities that are denoted by...
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 Fall '07
 Carlton

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