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1 T Bell, I H Witten, and J G Cleary, "Modelling for Text ComSurveys 21 ( 1989), 557-59 1 pression," ACM Comp~lting 2 R G Gallager, "Variations on a Theme by Huffman," IEEE Transactions on Information Theor IT-24 (1978), 668-674 3 D A Huffman, "A Model for the Construction of Minimum Redundancy Codes," Proceedings qf the IRE 40 ( 1952), 1098-1 101 4 D E Knuth, "Dynamic Huffman Coding," Jollrnal of Algorithms 6 (1985), 163-180 5 D A Lelewer and D S Hirschberg, "Data Compression," ACM Computing Surveys 19 ( 1 987) 26 1-296 6 T A Welch "A Technique for High-Performance Data Compres17 sion," Comp~iter (1984), 8-19 7 J Ziv and A Lempel, "Compression of Individual Sequences via Variable-Rate Coding," IEEE Transactions on Infornzatiorz Theor IT-24 (I 978), 530-536

14

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An important use of computers is for simulation, in which the computer is used to emulate the operation of a real system and gather statistics For example, we might want to simulate the operation of a bank with k tellers to determine the minimum value of k that gives reasonable service time Using a computer for this task has many advantages First the information would be gathered without involving real customers Second, a simulation by computer can be faster than the actual implementation because of the speed of the computer Third the simulation could be easily replicated In many cases, the proper choice of data structures can help us improve the efficiency of the simulation In this chapter, we show: how to simulate a game modeled on the Joseph~ls problern, and how to simulate the operation of a computer modem bank

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An important use of computers is simulation, in which the computer is used to emulate the operation of a real system and gather statistics

with pO == E, kO == "', and so on, and where in reaching the last line we have performed the d 3k integration using

The Josephus problem is the following game: N people, numbered 1 to N, are sitting in a circle; starting at person I , a hot potato is passed; after M passes, the person holding the hot potato is eliminated, the circle closes ranks and the game continues with the person who was sitting after the eliminated person picking up the hot potato; the last remaining person wins A common assumption is that M is a constant, although a random number generator can be used to change M after each elimination The Josephus problem arose in the first century 4 ~ in a cave on a mountain in Israel where Jewish zealots were being besieged by Roman soldiers The historian Josephus was among them To Josephus's consternation the zealots voted to enter into a suicide pact rather than surrender to the Romans He suggested the game that now bears his name The hot potato

In the Josephus problem, a hot potato is repeatedly passed; when passing terminates, the player holding the potato is eliminated; the game continues, and the last remaining player wins

The Josephus problem: At each step, the darkest circle represents the initial holder and the lightly shaded circle represents the player who receives the hot potato (and is eliminated) Passes are made clockwise

= fd4k8("')c5(k2).

Linearity. The essential feature of a linear regression function is that it is a sum of terms f3 j X j , where f3 j is an unknown parameter and Xj is a known value obtained from the data. The manner in which Xj is obtained from the data is not important. For example, suppose we have data on the ages (Y) and diameters (X) of pine trees. One linear regression model for these data is Y = f30 + f3IX + e. Another linear regression model we could try is Y = f30 + f31 X + f32 X 2 + e. The latter model might look like a quadratic model rather than a linear model because it involves X2. But if we simply change notation by setting XI = X and X 2 = X2, then we can write the model as Y = f3 0 + f31 X I + f32 X 2 + e, which falls under the definition of a linear regression model. The General Model. In general, suppose we have data consisting of n observations of a response variable Y and p explanatory variables XI' X 2 , , Xp. It is convenient to arrange the data in an array as in Table 1.2. Such data are sometimes called "regression data" because it is natural to think of using a regression model to analyze them. The linear regression model is the equation

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