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MTH/STA 561 RANDOM VARIABLES 1.10 Random Variables Up to now our discussion of probability theory has been concerned with situations in which the sample points of a sample space are arbitrary objects that may not be numbers. In the experiment of tossing a coin, for example, the outcome is either head or tail which is not a number. For many purposes, it is convenient to convert the outcome of an experiment into a numerical value, in particular to be able to use the familiar structure of the real numbers. Often we are not interested in the details associated with each sample point but only in some numerical description of the outcome. For instance, the sample space with a detailed description of each possible outcome of tossing a coin three times may be written S = fHHH; HHT; HT H; HT T; T HH; T HT; T T H; T T T; g : If we are concerned only with the number of heads that appear, then a numerical value of 0, 1, 2, or 3 will be assigned to each sample point. Thus, we are concerned only with experiments in which the outcomes either are numerical themselves or have a numerical value assigned to them. The sample space or the induced sample space (in the latter case) is then a space of numbers or a pace of vectors of numbers, and the structure of such spaces allows analyses and descriptions that may not be possible in the general case. The number of heads in the three tosses of a coin is referred to as a random variable, which is a random quantity determined, at least in part, by some chance mechanism on the outcome of the experiment. The most frequent use of probability theory is the description of random variables. We will study how random variables are de…ned and how to describe their behavior, and, subsequently, how to specify their probability laws to various ways. Suppose that we have an experiment associated with a sample space. Any random variable de…ned on the sample space can be referred to as a rule that associates a real number with each element in the sample space. In other words, a random variable is a function whose domain is the sample space and whose range is a set of real numbers. The formal de…nition may be stated as follows. De…nition 1. A random variable is a numerically valued function whose values correspond to the various outcomes of an experiment; that is, whose domain is the sample space. We shall use a capital letter, say Y , to denote a random variable and its corresponding lower case y to denote one of its values. Example 1. Consider an experiment of tossing a pair of fair coins. The sample space is S = fHH; HT; T H; T T g : 1 Let Y be the number of heads observed. Then we see that Sample Point HH HT TH TT Y 2 1 1 0 Thus, fY = 0g = fT T g ; fY = 1g = fHT; T Hg ; fY = 2g = fHHg : Hence, 1 4 2 1 P fY = 1g = = 4 2 1 P fY = 2g = : 4 P fY = 0g = Example 2. Consider an experiment of tossing three fair coins. The sample space is S = fHHH; HHT; HT H; HT T; T HH; T HT; T T H; T T T g : Let Y be the number of heads observed. We note that the random variable Y is a function from S into a set of nonnegative integers de…ned by Y (HHH) = 3 Y (T HH) = 2 Y (HHT ) = 2 Y (T HT ) = 1 Y (HT H) = 2 Y (T T H) = 1 Y (HT T ) = 1 Y (T T T ) = 0: Thus, each possible value of Y represents an event of the sample space S. Thus, fY fY fY fY = 0g = 1g = 2g = 3g = = = = fT T T g ; fHT T; T HT; T T Hg ; fHHT; HT H; T HHg ; fHHHg : Hence, 1 8 3 P fY = 1g = 8 3 P fY = 2g = 8 1 P fY = 3g = : 8 P fY = 0g = 2 Example 3. A urn contains 3 white balls and 4 red balls. We draw two balls in succession without replacement from the urn and let Y be the number of white balls drawn. Then the sample space is S = fW W; W R; RW; RRg : Since Y is the number of white balls drawn, we see that Sample Point WW WR RW RR Y 2 1 1 0 Thus, fY = 0g = fRRg fY = 1g = fW R; RW g fY = 2g = fW W g : Hence, P fY = 0g = P fRRg = 4 3 2 = 7 6 7 P fY = 1g = P fW Rg + P fRW g = P fY = 2g = P fW W g = 4 3 4 4 3 + = 7 6 7 6 7 3 2 1 = : 7 6 7 Example 4. A spinner can land in any of four positions, A, B, C, and D, with equal probability. The spinner is used twice and the position noted each time. Let the random variable Y denote the number of positions that the spinner did not land on. Compute the probabilities for each value of Y . Solution. The sample space is S = fAA; AB; AC; AD; BA; BB; BC; BD; CA; CB; CC; CD; DA; DB; DC; DDg : Since Y is the number of positions that the spinner did not land on, we see that Sample Point AA AB AC AD BA BB BC BD Sample Point CA CB CC CD DA DB DC DD Y 3 2 2 2 2 3 2 2 3 Y 2 2 3 2 2 2 2 3 Thus, fY = 2g = fAB; AC; AD; BA; BC; BD; CA; CB; CD; DA; DB; DCg fY = 3g = fAA; BB; CC; DDg : Hence, 12 3 = 16 4 4 1 P fY = 3g = = : 16 4 P fY = 2g = Example 5. Toss a pair of balanced dice and let Y be the sum of the the two numbers that appear. The sample space is S = f(y1 ; y2 ) j y1 = 1; 2; ; 6 and y2 = 1; 2; ; 6g : Then for (y1 ; y2 ) 2 S: Y (y1 ; y2 ) = y1 + y2 In particular, fY = 7g = f(1; 6) ; (2; 5) ; (3; 4) ; (4; 3) ; (5; 2) ; (6; 1)g and P fY = 7g = 6 1 = : 36 6 Example 6. A single dart is tossed at a bull’s-eye target by an experienced player. We are sure that the dart will hit somewhere within the outer ring, whose diameter is 8 inches. For the convenience of illustration, we superimpose an (y1 ; y2 ) rectangular coordinate system over the target, with its origin located at the center of the bull’s-eye. Then the sample space of the experiment de…ned by observing the impact point of one dart thrown by the player is given by S = (y1 ; y2 ) j y12 + y22 64 ; where the units used are inches and the impact point of the dart is recorded by its (y1 ; y2 ) coordinate (see Figure 1.6 ). If we de…ne Y to be the distance between the dart’s impact point and the center of the target (that is, the origin of p the rectangular coordinate system), then Y is a random variable that associates the number y12 + y22 with every element of S, that is, q Y (y1 ; y2 ) = y12 + y22 for (y1 ; y2 ) 2 S; p since the radial distance of any point (y1 ; y2 ) from the origin is y12 + y22 . For example, the radial distance of point (3; 4) from the origin is 5 as shown in Figure 1.6. 4 The range of random variables is the collection of real numbers associated with elements of the sample space S. For instance, the ranges of the random variables of the preceding …ve examples are, respectively, given by f0; 1; 2g f0; 1; 2; 3g f0; 1; 2g f2; 3g f2; 3; ; 4; ; 12g fyj 0 y 8g for for for for for for Example Example Example Example Example Example 1 2 3 4 5 6 The random variables in Examples 1 through 5 are discrete since their ranges are …nite sets, while the random variable in Example 6 is continuous since its range is an uncountable set. De…nition 2. A random variable Y is said to be discrete if its range is a …nite or countable set. Likewise, when a random variable assumes its values on a continuous scale (or on an uncountable set), it is called a continuous random variable. In most practical problems, discrete random variables represent count data, such as the number of defectives in a sample of n items or the number of phone calls per day in a certain type of switchboard, whereas continuous random variables represent measured data, such as all possible heights, weights, temperatures, distances, water levels of a river bank, or life periods. 5