Generate and test random numbers
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This document discusses methods for generating and testing random numbers. There are two main types of random number generators discussed: combined generators and inversive generators. Combined generators work by combining the outputs of two or more simpler random number generators. They are useful for simulating highly reliable systems or complex networks. The document also discusses how to test random numbers using the Kolmogorov-Smirnov test and runs tests. The Kolmogorov-Smirnov test compares the cumulative distribution function of observed values to expected values, while runs tests examine the arrangements of values in a sequence. Both can be used to determine if a random number generator is producing independent and identically distributed values.
![Generate and Test Random NumbersGenerate and Test Random NumbersExample:Two generators “k = 2”, a1 = 40014, m1 = 2147483563, a2 = 40692, m2 = 2147483399.Algorithm:Choose two seeds, X1,0 from [1, 2147483562] and X2,0 from [1, 2147483398], Set j = 0.Calculate the values from the two generators: Then calculate:After that return:Finally: j = j + 1, and then go back to step number 2.](https://image.slidesharecdn.com/generateandtestrandomnumbers-110113072543-phpapp01/85/Generate-and-test-random-numbers-8-320.jpg)
Generate and test random numbers
- 1. Generate and Test Random NumbersEng. MshariAlabdulkarim
- 2. Generate and Test Random NumbersGenerate and Test Random NumbersRandom Number Generation
- 3. Generate and Test Random NumbersGenerate and Test Random NumbersTypes of Random-number Generators:Combined generators.
- 4. Inversive Generators.Generate and Test Random NumbersGenerate and Test Random NumbersWhen to use Combined Generators :In the simulation of a highly reliable systems, where we need to simulate a huge number of events to observe any error.
- 5. In the simulation of a complex networks, where there are a huge number of users running lots of programs.Generate and Test Random NumbersGenerate and Test Random NumbersCombined Linear CongruentialGenerators :In 1988, L’Ecuyer suggested a new way to generate a sequence numbers with a very long period, and that by combining two or more congruential generators.
- 6. If Wi,1, Wi,2,..., Wi,kare independent, discrete-valued random variables, and Wi,1 is uniformly distributed between 0 and m1 – 2, then: is also uniformly distributed between 0 and m1 – 2.
- 7. Generate and Test Random NumbersGenerate and Test Random NumbersCombined Linear Congruential Generators (Cont):L’Ecuyer suggested to combined generator of the form:With:The maximum possible period will be:
- 8. Generate and Test Random NumbersGenerate and Test Random NumbersExample:Two generators “k = 2”, a1 = 40014, m1 = 2147483563, a2 = 40692, m2 = 2147483399.Algorithm:Choose two seeds, X1,0 from [1, 2147483562] and X2,0 from [1, 2147483398], Set j = 0.Calculate the values from the two generators: Then calculate:After that return:Finally: j = j + 1, and then go back to step number 2.
- 9. Generate and Test Random NumbersGenerate and Test Random NumbersExample (Cont.):Period:
- 10. Generate and Test Random NumbersGenerate and Test Random NumbersInversiveCongruential Generator :Invarsivecongruential generators are a type of nonlinear congruential pseudorandom number generator.
- 11. The standard formula for an inversivecongruential generator is:Generate and Test Random NumbersGenerate and Test Random NumbersTesting Random Number
- 12. Generate and Test Random NumbersGenerate and Test Random NumbersTypes of Random-number Testors:Kolmogorov-Smirnov Test.
- 13. Runs Tests.Generate and Test Random NumbersGenerate and Test Random NumbersKolmogorov-Smirnov Test :Developed by A. N. Kolmogorov and N. V. Smirnov.
- 14. Designed for continuous distributions.
- 15. Difference between the observed CDF (cumulative distribution function) Fo(x) and the expected cdf Fe(x) should be small.ObservedExpected
- 16. Generate and Test Random NumbersGenerate and Test Random NumbersKolmogorov-Smirnov Test :
- 17. Generate and Test Random NumbersGenerate and Test Random NumbersExample:
- 18. Generate and Test Random NumbersGenerate and Test Random NumbersExample (Cont.):
- 19. Generate and Test Random NumbersGenerate and Test Random NumbersExample (Cont.):
- 20. Generate and Test Random NumbersGenerate and Test Random NumbersExample (Cont.):
- 21. Generate and Test Random NumbersGenerate and Test Random NumbersRun Tests (Runs up and runs down):The runs test examines the arrangement of numbers in sequence to test the hypothesis of independence.
- 22. A run is defined as a succession of similar events preceded and followed by different event.
- 23. The length of the run is the number of events that occur in the run.
- 24. There are two Concerns in a runs test:Number of runs.Length of runs.
- 25. Generate and Test Random NumbersGenerate and Test Random NumbersRun Tests (Runs up and runs down):If N is the number of numbers in a sequence, the maximum number of runs is N-1, and the minimum number of runs is one.
- 26. If α is the total number of runs in a truly random sequence, then:
- 27. Mean:
- 28. Variance:
- 29. For N > 20, the distribution of “a” approximated by a normal distribution, N(ma , ).This approximation can be used to test the independence of numbers from a generator.
- 30. Generate and Test Random NumbersGenerate and Test Random NumbersRun Tests (Runs up and runs down):The standardized normal test statistic is developed by subtracting the mean from the observed number of runs “α”, and dividing by the standard deviation. That is, the test statistic is:
- 31. Failure to reject the hypothesis of independence occurs when: Where α is the level of significance.Fail to reject
- 32. Generate and Test Random NumbersGenerate and Test Random NumbersExample:Based on runs up and runs down, determine whether the following sequence of 40 numbers is such that the hypothesis of independence can be rejected where a = 0.05.
- 33. The sequence of runs up and down is as follows:+ + + -+-+- - - + + -+- - +-+- - +- - +-+ + - - + + -+- - + + -
- 34. Generate and Test Random NumbersGenerate and Test Random NumbersExample (Cont.):There are 26 runs in this sequence. With N=40 and a=26:
- 35. Now, the critical value is Z0.025 = 1.96, so the independence of the numbers cannot be rejected on the basis of this test.