used in probability modeling in Java

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EXAMPLE 171
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Throwing a Fair Die
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Let us suppose that we throw a fair die once Consider the function g(t) = (1/6)t + (1/6)t 2 + (1/6)t 3 + (1/6)t 4 + (1/6)t 5 + (1/6)t 6 and the random variable X denoting the face that appears The coef cient of t k in g(t) is P(X = k) for k = 1, 2, , 6 This is shown in Figure 171 Consider tossing the die twice with X1 and X2 , the relevant random variables Now calculate [g(t)]2 This is 1 2 [g(t)]2 = [t + 2t 3 + 3t 4 + 4t 5 + 5t 6 + 6t 7 + 5t 8 + 4t 9 + 3t 10 + 2t 11 + t 12 ] 36 The coef cient of t k is now P(X1 + X2 = k) A graph of this is interesting and is shown in Figure 172 This process can be continued, the coef cients of [g(t)]3 giving the probabilities associated with the sum when three fair dice are thrown The result is shown in Figure 173 This normal-like pattern continues Figure 174 shows the sums when 24 fair dice are thrown Since the coef cients represent probabilities, g[t] and its powers are called generating functions
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A Probability and Statistics Companion, John J Kinney Copyright 2009 by John Wiley & Sons, Inc
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03 025 02 015 01 005 1 2 3 X1 4 5 6
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004 Frequency 003 002 001 20 40 60 80 Sum 100 120
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MEANS AND VARIANCES
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This behavior of sums is a consequence of the central limit theorem, which states that the probability distribution of sums of independent random variables approaches a normal distribution If these summands, say X1 , X2 , X3 , , Xn , 2 2 2 2 have means 1 , 2 , 3 , , n and variances 1 , 2 , 3 , , n , then X1 + X2 + X3 + + Xn has expectation 1 + 2 + 3 + + n and variance 2 2 2 2 1 + 2 + 3 + + n Our example illustrates this nicely Each of the Xi s has the same uniform distribution with i = 7/2 and i2 = 35/12, i = 1, 2, 3, , n So we nd the following means and variances in Table 171 for various numbers of summands
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Table 171 n 1 2 3 24 7/2 7 21/2 84 2 35/12 35/6 35/4 70
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EXAMPLE 172
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Throwing a Loaded Die
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The summands in the central limit theorem need not all have the same mean or variance Suppose the die is loaded and the generating function is h(t) = t t2 t3 t4 t5 2t 6 + + + + + 10 5 20 20 5 5
A graph of this probability distribution, with variable X again, is shown is Figure 175 When we look at sums now, the normal-like behavior does not appear quite so soon Figure 176 shows the sum of three of the loaded dice But the normality does appear Figure 177 shows the sum of 24 dice
04 035 03 025 02 015 01 005 1 2 3 X1 4 5 6
Frequency
17
Generating Functions
015 0125 01 0075 005 0025 25 5 75 10 125 15 X1 + X2 +X3
Frequency
Frequency
003 002 001 20 40 60 80 Sum 100 120
The pattern for the mean and variances of the sums continues 1 = 17 and 24 = 102 4
2 while 1 =
279 837 2 and 24 = 80 10
A NORMAL APPROXIMATION
Let us return now to the case of the fair die and the graph of the sum of 24 tosses of the die as shown in Figure 174 Since E[X1 ] = 7/2 and Var[X1 ] = 35/12, we know that E[X1 + X2 + X3 + + X24 ] = 24 and Var[X1 + X2 + X3 + + X24 ] = 24 35 = 70 12 7 = 84 2
The visual evidence in Figure 174 suggests that the distribution becomes very normal-like so we compare the probabilities when 24 dice are thrown with values of a normal distribution with mean 84 and standard deviation 70 = 83666 A comparison is shown in Table 172 S denotes the sum So the normal approximation is excellent This is yet another illustration of the central limit theorem Normality, in fact, occurs whenever a result can be considered as the sum of independent random variables We nd that many human characteristics such as height, weight, and IQ are normally distributed If the measurement can be considered to be the result of the sum of factors, then the central limit theorem assures us that the result will be normal
Explorations Table 172 S 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 Probability 00172423 00202872 00235250 00268886 00302958 00336519 00368540 00397959 00423735 00444911 00460669 00470386 0047367 00470386 00460669 00444911 00423735 00397959 00368540 00336519 00302958 00268886 00235250 00202872 Normal 00170474 00200912 00233427 00267356 00301874 00336014 0036871 00398849 00425331 00447139 00463396 00473433 00476827 00473433 00463396 00447139 00425331 00398849 0036871 00336014 00301874 00267356 00233427 00200912