1. Random Numbers

2. Two Types of Random Numbers 1.True random numbers: True random numbers are generated in non- deterministic ways. They are not predictable. They are not repeatable. 2.Pseudorandom numbers: Pseudorandom numbers are numbers that appear random, but are obtained in a deterministic, repeatable, and predictable manner.

3. True Random Numbers Generators  Use one of several sources of randomness –decay times of radioactive material –electrical noise from a resistor or semiconductor –radio channel or audible noise –keyboard timings  some are better than others  usually slower than PRNGs

4. Pseudorandom Numbers Generators Reasons for pseudorandom numbers: –Flexible policies –Lack of knowledge Generate stochastic processes Decision making (random decision) Numerical analysis (numerical integration) Monte Carlo integration

5. Pseudorandom Numbers Generators The desirable Properties of Pseudorandom Numbers –Uncorrelated Sequences - The sequences of random numbers should be serially uncorrelated –Long Period - The generator should be of long period (ideally, the generator should not repeat; practically, the repetition should occur only after the generation of a very large set of random numbers). –Uniformity - The sequence of random numbers should be uniform, and unbiased. That is, equal fractions of random numbers should fall into equal ``areas'' in space. Eg. if random numbers on [0,1) are to be generated, it would be poor practice were more than half to fall into [0, 0.1), presuming the sample size is sufficiently large. –Efficiency - The generator should be efficient. Low overhead for massively parallel computations.

6. How to generate pseudorandom random numbers Random number seed: Virtually all computer methods of random number generation start with an initial random number seed. This seed is used to generate the next random number and then is transformed into a new seed value.

7. Midsquare Method 1.Start with an initial seed (e.g. a 4-digit integer). 2.Square the number. 3.Take the middle 4 digits. 4.This value becomes the new seed. Divide the number by 10,000. This becomes the random number. Go to 2.

8. Midsquare Method, example x 0 = 5497 x 1 : 5497 2 = 30217009  x 1 = 2170, R 1 = 0.2170 x 2 : 2170 2 = 04708900  x 2 = 7089, R 2 = 0.7089 x 3 : 7089 2 = 50253921  x 3 = 2539, R 3 = 0.2539 Drawback: Hard to state conditions for picking initial seed that will generate a “good” sequence.

9. Midsquare Generator, examples “Bad” sequences: x 0 = 5197 x 1 : 5197 2 = 27008809  x 1 = 0088, R 1 = 0.0088 x 2 : 0088 2 = 00007744  x 2 = 0077, R 2 = 0.0077 x 3 : 0077 2 = 00005929  x 3 = 0059, R 3 = 0.0059 x i = 6500 x i+1 : 6500 2 =42250000  x i+1 =2500, R i+1 = 0.2500 x i+2 : 2500 2 =06250000  x i+2 =2500, R i+1 = 0.2500

10. Linear Congruential Generator (LCG) Generator Start with random seed Z 0 < m = largest possible integer on machine Recursively generate integers between 0 and M Z i = (a Z i-1 + c) mod m Use U = Z/m for pseudo-random number get (avoid 0 and 1) When c = 0  Called Multiplicative Congruential Generator When c > 0  Mixed LCG

11. Linear Congruential Generator (LCG) (Lehmer 1951) Let Z i be the i th number (integer) in the sequence Z i = (aZ i-1 +c)mod(m) Z i  {0,1,2,…,m-1} where Z 0 = seed a = multiplier c = increment m = modulus DefineU i = Z i /m (to obtain U(0,1) value)

12. LCG, example 16-bit machine a = 1217c = 0 Z 0 = 23m = 2 15 -1 = 32767 Z 1 = (1217*23) mod 32767 = 27991 U 1 = 27991/32767 = 0.85424 Z 2 = (1217*27991) mod 32767 = 20134 U 2 = 20134/32767 = 0.61446

13. An LCG can be expressed as a function of the seed Z 0 THEOREM: Z i = [a i Z 0 +c(a i -1)/(a-1)] mod(m) Proof:By induction on i i=0Z 0 = [a 0 Z 0 +c(a 0 -1)/(a-1)] mod(m) Assume for i. Show that expression holds for i+1 Z i+1 = [aZ i +c] mod(m) = [a {[a i Z 0 +c(a i -1)/(a-1)] mod(m)} +c] mod(m) = [a i+1 Z 0 +ac(a i -1)/(a-1) +c] mod(m) = [a i+1 Z 0 +c(a i+1 -1)/(a-1) ] mod(m)

14. Examples: Z i = (69069Z i-1 +1) mod 2 32 U i = Z i /2 32 Z i = (65539Z i-1 +76) mod 2 31 U i = Z i /2 31 Z i = (630360016Z i-1 ) mod (2 31 -1)U i = Z i /2 31 Z i = 13 13 Z i-1 mod 2 59 U i = Z i /2 59 What makes one LCG better than another?

15. Mixed congruential generator is full period if 1.m = 2 B (B is often # bits in word) fast 2.c and m relatively prime (g.c.d. = 1) 3.If 4 divides m, then 4 divides a – 1 (e.g., a = 1, 5, 9, 13,…)

16. A full period (full cycle) LCG generates all m values before it cycles. Consider Z i = (3Z i-1 +2) mod(9) with Z 0 =7 Then Z 1 = 5 Z 2 = 8 Z 3 = 8 Z j = 8 j = 3,4,5,6,… On the other hand Z i = (4Z i-1 +2) mod(9) has full period. Why?

17. The period of an LCG is m (full period or full cycle) if and only if —If q is a prime that divides m, then q divides a-1 —The only positive integer that divides both m and c is 1 —If 4 divides m, then 4 divides a-1. Examples Z i+1 = (16807Z i +3) mod (451605), where 16807 =7 5, 16806 =(2)(3)(2801), 451605 =(3)(5)(7)(11)(17)(23) This LCG does not satisfy the first two conditions. Z i+1 = (16807Z i +5) mod (635493681) where 16807 =7 5, 16806 = (2)(3)(2801), 635493681 = (3 4 )(2801 2 ) This LCG satisfies all three conditions.

18. - m = 2 B where B = # bits in the machine is often a good choice to maximize the period. - If c = 0, we have a power residue or multiplicative generator. Note that Z n = (aZ n-1 ) mod(m)  Z n = (a n Z 0 ) mod(m). If m = 2 B, where B = # bits in the machine, the longest period is m/4 (best one can do) if and only if —Z 0 is odd —a = 8k+ 3, k  Z + (5,11,13,19,21,27,…)

19. Lagged Fibonacci Generators  Similar to Fibonacci Sequence  Increasingly popular  X n = (X n-l + X n-k ) mod m (l>k>0)  l seeds are needed  m usually a power of 2  Maximum period of (2 l -1)x2 M-1 when m=2 M

20. Add-with-carry & Subtract-with-borrow  Similar to LFG  AWC: X n =(X n-l +X n-k +carry) mod m  SWB: X n =(X n-l -X n-k -carry) mod m

21. Multiply-with-carry Generators  Similar to LCG  X n =(aX n-1 +carry) mod m

22. Random Number Generation Other kinds of generators Quadratic Congruential Generator –S new = (a 1 S old 2 + a 2 S old 2 + b) mod L Combination of Generators –Shuffling – L’Ecuyer – Wichman/Hill Tausworthe Generator –Generates sequence of random bits

23. Assignment Generate 20 Pseudorandom number using: 1.Midsquare Method 2.Multiplicative Linear Congruential Generator (LCG) Generator 3.Mixed Linear Congruential Generator (LCG) Generator