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# chap05 - Chapter 5 Introduction to Random Variables...

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Chapter 5 Introduction to Random Variables Consider a random experiment described by a triplet ( S, F ,P ). In applica- tions of probabilities, we are often interested in numerical quantities derived from the experimental outcomes. These quantities may be viewed as func- tions from the sample space S into the set of real numbers R , as in: s S X ( s ) R Provided certain basic requirements are satisﬁed, these quantities are gener- ally called random variables . As an example, consider the sum of the two numbers showing up when rolling a fair die twice: The set of all possible outcomes is S = { ( i,j ) : ∈ { 1 , 2 ,..., 6 }} . The sum of the two numbers showing up may be represented by the functional relationship s = ( ) X ( s ) = i + j. Note that the function X ( s ) may be used in turn to deﬁne more complex events. For instance, the event that the sum is greater or equal to 11 may be expressed concisely as A = { s S : X ( s ) 11 } 117

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118 The terminology random variable is appropriate in this type of situations because: The value X ( s ) depends on the experimental outcome s . The outcome s of a particular realization of the random experiment (i.e. a trial) is unknown beforehand, and so is X ( s ). Each experimental trial may lead to a diﬀerent value of X ( s ) Random variables are extremely important in engineering applications. They are often used to model physical quantities of interest that cannot be pre- dicted exactly due to uncertainties. Some examples include: Voltage and current measurements in an electronic circuit. Number of erroneous bits per second in a digital transmission. Instantaneous background noise amplitude at the output of an audio ampliﬁer. Modelization of such quantities as random variables allows the use of proba- bility in the design and analysis of these systems. This and the next few Chapters are devoted to the study of random variables, including: deﬁnition, characterization, standard models, properties, and a lot more. .. In this Chapter, we give a formal deﬁnition of a random variable, we introduce the concept of a cumulative distribution function and we introduce the basic types of random variables. c ± 2003 Benoˆ ıt Champagne Compiled February 10, 2012
5.1 Preliminary notions 119 5.1 Preliminary notions Function from S into R : Let S denote a sample space of interest. A function from S into R is a mapping, say X , that associate to every outcome in S a unique real number X ( s ): S mapping X real axis s 1 s 2 s 3 X(s 1 ) X(s 2 )=X(s 3 ) Figure 5.1: Illustration of a mapping X from S into R . The following notation is often used to convey this idea: X : s S X ( s ) R . (5.1) We refer to the sample space S as the domain of the function X . The range of the function X , denoted R X , is deﬁned as R X = { X ( s ) : s S } ⊆ R (5.2) That is, R X is the of all possible values for X ( s ), or equivalently, the set of all real numbers that can be “reached” by the mapping X .

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chap05 - Chapter 5 Introduction to Random Variables...

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