1. Generate and Test Random Numbers Eng. MshariAlabdulkarim

2. Generate and Test Random Numbers Generate and Test Random Numbers Random Number Generation

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.

8. Generate and Test Random Numbers Generate and Test Random Numbers Example: 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 Numbers Generate and Test Random Numbers Example (Cont.): Period:

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.Observed Expected

16. Generate and Test Random Numbers Generate and Test Random Numbers Kolmogorov-Smirnov Test :

17. Generate and Test Random Numbers Generate and Test Random Numbers Example:

18. Generate and Test Random Numbers Generate and Test Random Numbers Example (Cont.):

19. Generate and Test Random Numbers Generate and Test Random Numbers Example (Cont.):

20. Generate and Test Random Numbers Generate and Test Random Numbers Example (Cont.):

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.

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.

31. Failure to reject the hypothesis of independence occurs when: Where α is the level of significance. Fail to reject

33. The sequence of runs up and down is as follows:+ + + -+-+- - - + + -+- - +-+- - +- - +-+ + - - + + -+- - + + -

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.