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- 1. Generate and Test Random Numbers<br />Eng. MshariAlabdulkarim<br />
- 2. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Random Number Generation<br />
- 3. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Types of Random-number Generators:<br /><ul><li>Combined generators.
- 4. Inversive Generators.</li></li></ul><li>Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />When to use Combined Generators :<br /><ul><li>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.</li></li></ul><li>Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Combined Linear CongruentialGenerators :<br /><ul><li>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:</li></ul> is also uniformly distributed between 0 and m1 – 2.<br />
- 7. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Combined Linear Congruential Generators (Cont):<br /><ul><li>L’Ecuyer suggested to combined generator of the form:</li></ul>With:<br />The maximum possible period will be:<br />
- 8. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example:<br />Two generators “k = 2”, a1 = 40014, m1 = 2147483563, a2 = 40692, m2 = 2147483399.<br />Algorithm:<br />Choose two seeds, X1,0 from [1, 2147483562] and X2,0 from [1, 2147483398], Set j = 0.<br />Calculate the values from the two generators:<br /> Then calculate:<br />After that return:<br />Finally:<br /> j = j + 1, and then go back to step number 2.<br />
- 9. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example (Cont.):<br />Period:<br />
- 10. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />InversiveCongruential Generator :<br /><ul><li>Invarsivecongruential generators are a type of nonlinear congruential pseudorandom number generator.
- 11. The standard formula for an inversivecongruential generator is:</li></li></ul><li>Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Testing Random Number<br />
- 12. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Types of Random-number Testors:<br /><ul><li>Kolmogorov-Smirnov Test.
- 13. Runs Tests.</li></li></ul><li>Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Kolmogorov-Smirnov Test :<br /><ul><li>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.</li></ul>Observed<br />Expected<br />
- 16. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Kolmogorov-Smirnov Test :<br />
- 17. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example:<br />
- 18. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example (Cont.):<br />
- 19. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example (Cont.):<br />
- 20. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example (Cont.):<br />
- 21. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Run Tests (Runs up and runs down):<br /><ul><li>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:</li></ul>Number of runs.<br />Length of runs.<br />
- 25. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Run Tests (Runs up and runs down):<br /><ul><li>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 , ).</li></ul>This approximation can be used to test the independence of numbers from a generator.<br />
- 30. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Run Tests (Runs up and runs down):<br /><ul><li>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:</li></ul> Where α is the level of significance.<br />Fail to reject<br />
- 32. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example:<br /><ul><li>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:</li></ul>+ + + -+-+- - - + + -+- - +-+- - +- - +-+ + - - + + -+- - + + -<br />
- 34. Generate and Test Random Numbers<br />Generate and Test Random Numbers<br />Example (Cont.):<br /><ul><li>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.</li>

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