1. Randomness Test Fall 2012 By Yaohang Li, Ph.D.

2. Review Last Class –Random Number Generation –Uniform Distribution This Class –Test of Randomness –Chi Square Test –K-S Test –10 empirical tests Next Class –Nuclear Simulation

3. Chi-square test Introduced by Karl Pearson in 1900 Test for discrete distributions e.g. binomial and Poisson distributions Implementation: - assume we have k possible categories - P = sequence size / k - expected sample size = P * n trials - suppose category i occurs Y i times - error = Y i – nP i - chi-square statistic - chi-square percentile = proportion of samples from a "true“ distribution having a chi-square statistic (function of errors) less than the percentile.

4. Example Given two “true” dice for 144 trials we get: s = 2 3 4 5 6 7 8 9 10 11 12 P s = 1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36 Y s = 2 4 10 12 22 29 21 15 14 9 6 nP s = 4 8 12 16 20 24 20 16 12 8 4 V = (Y 2 – nP 2 )² / nP 2 + (Y 3 – nP 3 )² / nP 3 +………+ (Y 12 – nP 12 )² / nP 12 V = (2 – 4)² / 4 + (4 – 8)² / 8 +……+ (9 – 8)² / 8 + (6 – 4)² / 4 = 7 7/48

5. Chi-square Table

6. Kolmogorov-Smirnov test Introduced in 1933 Test for continuous distributions e.g. normal and Weibull distributions based on ECDF defined as,

7. K-S Test

8. K-S Table

9. Empirical Tests Equidistribution Test Serial Test Gap Test Poker Test Coupon Collector’s Test Permutation Test Run Test Maximum of t test Collision Test Serial Correlation Test

10. Summary Chi-Square Test KS Test Empirical Tests

11. What I want you to do? Review Slides Review basic probability/statistics concepts Work on your Assignment 3