RANDOM NUMBER GENERATION

1
RANDOM NUMBER GENERATION

• Week 3
• Kelton text (Ch. 12)
• GE-703 Kumpaty

2
Generating Random Numbers

• LCG (linear congruential generators) Zi
  (a Zi-1 c) mod (m)
• Mixed (above) Multiplicative when c0.
• Seed value Z0
• Maximum cycle length achieved is m.
• Promodel uses a630360016 c0 and m231-1 prime
  modulus multiplicative LCG
• Arena (used to have same m, c, a16807)
• Cycle length in 100 unique streams

3
Generating Random Numbers

• Zi (a Zi-1 c) mod (m) Guidelines for selection
  of m,a and c.

4
Random Number Generators

• Will the following LCG work?
• Zi (12Zi-15) mod (32) with Z0 29
• i Zi
• 0 29
• 1 1
• 2 17
• 3 17
• 4 17
• 5 17
• It wont work because a4k1 is violated.

5
Testing Random Number Generators

• 1. Testing for independence
• Hypothesis
• H0 Ui values from the generator are
  independent (Ui for uniform distr. Zi/m)
• H1 Ui values from the generator are not
  independent
• Specified significance level a
• The Runs test
• The Runs Above and Below the Median Test
• The Runs Up and Runs Down Test

6
Testing Random Number Generators

• 2. Testing for uniformity/homogeneity
• Hypothesis
• H0 Ui values are uniform
• H1 Ui values are not uniform
• Specified significance level a
• The Kolmogorov-Smirnov test
• The Chi-Square test
• Will do in Chapter 6

7
Generating Random Variates

• Inverse Transformation Method
• CONTINOUS Distributions
• f(x) probability distribution function
• F(x) cumulative distribution function, F(x)
  P(Xltx)
• Set U F(x) where U is uniform (0,1) and solve
  for x.
• x F-1(U) x is the random variate

8
Cumulative Distribution function for Uniform
Distribution

• Pr (0 ? x ? x0) F(x0) a? x0 1/(b-a) dx
• F(x0) (x0-a)/(b-a)
• Also if the cumulative probability F(x0) is known
  we can also determine x0 as
• x0 a(b-a) F(x0)
• Example b7 a4 given F(x0)0.53, x0 5.59

9
Cumulative Distribution function for Exponential
Distribution

• Pr (0 ? x ? x0) F(x0) 0? x0 b e-bx dx
• F(x0) - e-bx 0x0 1 - e-bx0
• Also if the cumulative probability F(x0) is known
  we can also determine x0 as
• x0 (-1/b) ln(1 F(x0))
• Please note (1/b) is the mean of the exponential
  distribution.