1. 1 Simulation Modeling and Analysis Pseudo-Random Numbers

2. 2 Outline Properties of Random Numbers Generating Random Numbers Testing Random Numbers

3. 3 Properties of Random Numbers Key Properties –Uniformity –Independence Density function (continuous!) f(x) = {1 for 0 < x < 1, 0 otherwise Moments E(R) = 1/2 V(R) = 1/12

4. 4 Generating Random Numbers Random Numbers vs Pseudo-random Numbers Requirements of a RNG routine –Speed –Portable –Long Cycle –Replicable RN –Uniform and Independent RN’s

5. 5 Random Number Generation Linear Congruential Method X i+1 = (a X i + c) mod m R i = X i /m Note: Only values from the set I = {0,1/m,2/m,…,(m-1)/m} are obtained

6. 6 Random Number Generation -contd Longest Possible Period (P) –If m = 2 b and |c| > 0, P = m –If m = 2 b and c = 0, P = m/4 –If m = prime and c = 0, P = m-1 Example: X i+1 = (7 5 X i ) mod (2 31 -1)

7. 7 Random Number Generation -contd Combined Congruential Generators. Two distinct congruential generators can be combined to obtain PRN’s with longer periods. X i+1 = (  (-1) j-1 X i,j ) mod (m 1 - 1) R i = X i /m 1, X i > 0 ; R i = (m 1 -1)/m 1, X i > 0

8. 8 Testing Random Numbers Null Hypotheses H 0 : R i ~ U[0,1] ; H 0 : R i ~ independent Tests –Frequency test –Runs test –Autocorrelation test –Gap test –Poker test

9. 9 Kolmogorov-Smirnov Frequency Test 1.- Arrange data in increasing value 2.- Compute D +, D - and D 3.- Find critical D c (Handout) for given a 4.- Accept or reject the null hypothesis. 5.- Example: Stat::Fit

10. 10 Chi-Square Frequency Test The Chi static compares observed frequencies of occurrence of PRN’s in selected subdomains against expected frequencies derived from the U distribution function. See Stat::Fit X 0 2 =  n (O i - E i ) 2 /E i

11. 11 Runs Testing Run: sequence of similar events Runs up and runs down (independence) –Maximum number of runs (N numbers) = N-1 –mean = (2N-1)/3; variance = (16N-29)/90 –Test hypothesis against normal distribution.

12. 12 Runs Testing -contd Runs above and below the mean –Maximum number of runs (N numbers, n1 above and n2 below the mean) = n1+n2 –mean = 2 n1 n2/N + 1/2 –variance = 2 n1 n2 (2 n1 n2 - N)/N 2 (N-1) –Test hypothesis against normal distribution.

13. 13 Runs Testing -contd Runs length –Test hypothesis against Chi square distribution

14. 14 Autocorrelation Testing Seek the autocorrelation between every m numbers (I.e. dependence) Null Hypothesis H 0 :  im = 0 Note: If values are uncorrelated,  im has normal distribution. So, test hypothesis against normal distribution.

15. 15 Gap Testing Gap: Interval of recurrence of same digit. Monitor Frequency of gaps and test 1.- Specify the cdf F(x) = 1-0.9 x+1 2.- Arrange the observed gaps into S(x) 3.- Find D and Dc 4.- Accept or reject the null hypothesis.

16. 16 Poker Test Frequency of repetition of certain digits in a series Null hypothesis is tested againts the Chi- square distribution.