Chap07Slides - 7-1 CHAPTER 7 Random-Number Generators 7.1

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Unformatted text preview: 7-1 CHAPTER 7 Random-Number Generators 7.1 Introduction................................................................................................2 7.2 Linear Congruential Generators ....................................................................6 7.3 Other Kinds of Generators.........................................................................11 7.3.1 More General Congruences...............................................................................................11 7.3.2 Composite Generators ......................................................................................................12 7.3.3 Tausworthe and Related Generators..................................................................................14 7.4 Testing Random-Number Generators .........................................................15 7.4.1 Empirical Tests..................................................................................................................16 7.4.2 Theoretical Tests...............................................................................................................17 7.4.3 Some General Observations on Testing..............................................................................20 7-2 7.1 Introduction The GoalAll stochastic simulations need to generate IID U(0,1) random numbers somehow Density function: =otherwise1if1)(xxfReason: Observations on allother RVs/processes require U(0,1) input f(x) 1 x 0 1 7-3 Early MethodsPhysical Cast lots Dice Cards Urns Shewhart quality-control methods (normal bowl) Students experiments on distribution of sample correlation coefficient Lotteries Mechanical Spinning disks (Kendall/Babington-Smith, 10,000 digits) Electrical ERNIE RAND Corp. Tables: A Million Random Digits with 100,000 Normal DeviatesOther schemes Pick digits randomly from Scottish phone directory or census reports Decimals in expansion of pto 100,000 places 7-4 Algorithmic, Sequential Computer MethodsSequential: the next random number is determined by one or several of its predecessors according to a fixed mathematical formula The midsquare method: von Neumann and Metropolis, 1945 Start with Z= 4-digit positive integer Z1= middle 4 digits of Z2(append 0s if necessary to left of Z2 to get exactly 8 digits); U1= Z1, with decimal point at left Z2= middle 4 digits of Z12; U2= Z2, with decimal point at left Z3= middle 4 digits of Z22; U3= Z3, with decimal point at left. etc. i ZiUiZi20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . .7182 5811 7677 9363 6657 3156 9603 2176 7349 0078 0060 0036 0012 0001 0000 0000 . . 0.5811 0.7677 0.9363 0.6657 0.3156 0.9603 0.2176 0.7349 0.0078 0.0060 0.0036 0.0012 0.0001 0.0000 0.0000 ....
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Chap07Slides - 7-1 CHAPTER 7 Random-Number Generators 7.1

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