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ECSE 305 (5)
Lecture

chap05.pdf

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Department
Electrical Engineering
Course Code
ECSE 305
Professor
Ioannis P.

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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 2 S ! X(s) 2 R Provided certain basic requirements are satis▯ed, 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 = f(i;j) : i;j 2 f1;2;:::;6gg. ▯ The sum of the two numbers showing up may be represented by the functional relationship s = (i;j) ! X(s) = i + j: ▯ Note that the function X(s) may be used in turn to de▯ne more complex events. For instance, the event that the sum is greater or equal to 11 may be expressed concisely as A = fs 2 S : X(s) ▯ 11g 117 118 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 di▯erent 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 ampli▯er. 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: de▯nition, characterization, standard models, properties, and a lot more... In this Chapter, we give a formal de▯nition of a random variable, we introduce the concept of a cumulative distribution function and we introduce the basic types of random variables. 2003 Beno^▯t 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): mapping X S s 3 s2 real axis s1 X(s1) X(s2)=X(s3) Figure 5.1: Illustration of a mapping X from S into R. ▯ The following notation is often used to convey this idea: X : s 2 S ! X(s) 2 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 , iX de▯ned as R X fX(s) : s 2 Sg ▯ 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. 2003 Beno^▯t Champagne Compiled February 10, 2012 5.1 Preliminary notions 120 Inverse function: ▯ Let X be a function from S into R. ▯1 ▯ We de▯ne the inverse function X as follows: for any subset D of R, ▯1 X (D) = fs 2 S : X(s) 2 Dg (5.3) ▯1 ▯ That is, X (D) is the subset of S containing all the outcomes s (possibly more than one) such that X(s) is in D. This is illustrated below. mapping X -1 S real axis X (D) D=[a,b] Figure 5.2: Illustration of inverse mapping X ▯ This de▯nition of an inverse is very general; it applies even in the case when X is many-to-one. Left semi-in▯nite intervals: ▯ To every x 2 R, we associate a left semi-in▯nite interval I x de▯ned as Ix, (▯1;x] = fy 2 R : y ▯ xg (5.4) ▯ The inverse image of I xnder mapping X is given by X ▯1 (Ix) = fs 2 S : X(s) ▯ xg (5.5) 2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 121 5.2 De▯nition of a random variable De▯nition: Let (S;F;P) be a probability space. A function X : s 2 S ! X(s) 2 R is called a random variable (RV) if X ▯1(I ) = fs 2 S : X(s) ▯ xg 2 F; for all x 2 R (5.6) x Discussion: ▯ According to this de▯nition, X de▯nes a mapping from sample space S into R, as illustrated in Figure 5.1 ▯ However, X is not arbitrary: we require that for any x 2 R, the inverse ▯1 image X (Ix), as illustrated in Figure 5.3, be a valid event. mapping X -1 S Ix real axis X (I ) x x Figure 5.3: Inverse image X(I ) x ▯ This condition ensures that P(fs 2 S : X(s) ▯ xg), i.e. the probability that X(s) belong to the interval I , xs well-de▯ned. 2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 122 Example 5.1: I A fair coin is ipped twice. Let random variable X represent the number of tails observed in this experiment. Here, the sample space may be de▯ned as S = fHH;HT;TH;TTg Since this is a ▯nite set, a proper choice of event algebra is F = P S f;;fHHg;:::;Sg 4 Note that F contains 2 = 16 events. According to the problem statement, the function X : S ! R may be computed as follows: s = HH ! X(s) = 0 s = HT or TH ! X(s) = 1 s = TT ! X(s) = 2 so that its range isXR= f0;1;2g. This is illustrated below: For any x 2 R, we have ▯1 X (x ) = fs 2 S : X(s) ▯ xg 2 F since x is a subset of S and F = S contains all the possible subsets of S. This shows that X is a valid random variable. For instance: x < 0 ) X ▯1(x ) = ; 2 F ▯1 0 ▯ x < 1 ) X (x ) = fHHg 2 F ▯1 1 ▯ x < 2 ) X (x ) = fHH;HT;THg 2 F ▯1 2 ▯ x ) X (x ) = S 2 F ▯ J 2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 123 Remarks on condition (5.6): ▯ By de▯nition, X ▯1(I ) = fs 2 S : X(s) ▯ xg is a subset of S. Therefore, x whenever F = P , cSndition (5.6) is satis▯ed and we need not worry about it. This is the case when S is ▯nite or countably in▯nite. ▯ When S is uncountably in▯nite and F = B , theSe exist functions X : S ! R that do not satisfy condition (5.6) and for which P(fs 2 S : X(s) ▯ xg) is not de▯ned. It is precisely to avoid this situation that (5.6) is included in the de▯nition of a random variable. ▯ In applications, we will want to compute probabilities of the type P(fs 2 S : X(s) 2 Dg) where D ▯ R represents any practical subset of real numbers. This includes, for example, intervals of the type [a;b], for any a ▯ b 2 R, as well as unions, intersections and/or complements of such intervals. ▯ As a consequence of condition (5.6), it can be shown that fs 2 S : X(s) 2 Dg) 2 F for any D 2 B ,Rso that P(fs 2 S : X(s) 2 Dg) is well-de▯ned for any practical subset of real-numbers (see next page for more on this). ▯ In this course, we shall work with relatively simple functions X and shall ▯1 always assume that the condition X (Ix) 2 F is satis▯ed. 2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 124 Further remarks (optional reading): ▯1 ▯ The condition X (Ix) 2 F in the de▯nition of random variable X ensures that ▯1 X (x ) = fs 2 S : X(s) ▯ xg ▯1 is a valid event, for which the probability P(X(Ix)) is well-de▯ned. ▯ More importantly, the condition ensures that for practical subsets of real numbers encountered in applications of probability, i.e. for any D 2RB , the Borel ▯eld of R, the set of experimental outcomes ▯1 X (D) = fs 2 S : X(s) 2 Dg is also a valid event for which a probability can be computed. ▯ While the detail of the proof are beyond the scope of this course, the justi▯cation of this statement involves three basic steps: - Any real number subset D 2 B R can be expressed as a combination of unions, intersections and complements of basic intervals of the typexI . ▯1 - Because X (Ix) 2 F for any x and because F is a ▯-algebra (closed under union, ▯1 intersection and complement), it follows that X (D) is also in F. ▯1 ▯1 - Finally, X (D) 2 F implies that P(X (D)) is well-de▯ned. ▯1 ▯ In the next section, we will see how P(X (Ix)) can actually be used in the computation ▯1 of P(X (D)). 2003 Beno^ ▯t Champagne Compiled February 10, 2012 5.
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