1. CS 450 – Modeling and Simulation Dr. X

2. Topics What Does Randomness Mean? Randomness in games Generating Random Values Random events in real life: measuring randomness

3. What is Randomness? Unpredictability? Does it characterize a single event or a sequence of events? From Wikipedia: Randomness means lack of pattern or predictability in events. Randomness suggests a non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination.ordercoherencesymbolssteps

4. Randomness in games

5. Probability 101 If you roll the dice twice, what is the probability of rolling two sixes?

6. Probability vs odds Odds = Favorable Outcomes / Unfavorable Outcomes Probability = Favorable Outcomes / All Outcomes

7. Probability 101 If you toss two coins what is the probability of HH?

8. Probability 101 Independent events Logical relations: – AND – OR

9. Defining Randomness How can a computer generate random numbers? Can we measure randomness?

10. Measuring Randomness Runs Test Frequency test Autocorrelation test Chi-square goodness of fit

11. Statistical Hypothesis test H 0 : Null Hypothesis, ex., a sequence of numbers is random H α : Alternative Hypothesis, ex., a sequence of numbers is NOT random Real Situation Decision H 0 is not rejectedH 0 is rejected H 0 is trueValidType I Error H 0 is not trueType II ErrorValid

12. Runs test Sequence of numbers X 1, …, X N Subtract X i+1 – X i Collect only the signs of the subtraction results Do the signs alternate ~50% of the time?

13. Frequency Test (Monobit test) Create a string S n = X 1 + X 2 + … + X n Calculate S obs = S n /n Calculate P-Value: P-value = erfc(S obs /2) where erfc: complementary error function If P-value > 0.01 accept the null hypothesis else reject

14. Autocorrelation Test Intuition: if the sequence of bits in e is random, then it will be different from another bit string obtained by shifting the bits of e by d positions.

15. Chi-Square Goodness of Fit Test Intuition: test checks whether a sequence of pseudo-random numbers in [0,1] are uniformly distributed If you have n numbers and split them in k groups, then if in the groups you have n/k numbers the distribution is normal Compare all observed values f i inside your kth group with the mean n/k If chi-square > theoretical x 2 obtained from table with k degrees of freedom, then null hypothesis rejected

16. Generating Random Numbers Congruential method Tausworthe generators Lagged Fibonacci generators The Mercenne Twister

17. Congruential Method x i+1 = ax i + c (mod m) a, c, m: non-negative If m is large, period of number repetition is large Conditions for full period: – m and c have no common divisor – a = 1 (mod r) if r is a prime factor of m – a = 1 (mod 4) if m is a multiple of 4

18. Tausworthe generators Additive congruential generators Modulus m is equal to 2 x i = (a 1 x i-1 + a 2 x i-2 +...+ a n x i-n ) (mod 2) where x i : 0 or 1 Mod 2 is equivalent to XOR

19. Lagged Fibonacci Generators Based on Fibonacci series: x n = x n-1 + x n-2 LFG: x n = x n-j O x n-k (mod m) where O: addition, or subtraction, or multiplication, or XOR

20. Mersenne Twister x k+n = x k+m ⊕ (x k u | x k+1 k )A, k≥0 Where x is the block of numbers of w bits

21. References The Guide to Computer Simulations and Games, By Katrin Becker and J.R.Parker, Wiley (Chapter 6) Computer Simulation Techniques: The Definitive Introduction, by Harry Perros, http://www4.ncsu.edu/~hp/simulation.pdf (Chapter 2) http://www4.ncsu.edu/~hp/simulation.pdf CS 45021