IV' Random Numbers

1
IV. Random Numbers

• M. Peter Jurkat
• CS452/Mgt532 Simulation for Managerial Decisions
• The Robert O. Anderson Schools of Management
• University of New Mexico

2
Need for Random Numbers

• To generate values from a statistical
  distribution
• Examples a time interval after the previous
  arrival (e.g., neg exp), a value of a service
  time (e.g., uniform), a number of items in an
  order (e.g., Poisson)
• Generation of random numbers satisfied this need
• Informally, random numbers are selections from a
  statistical distribution

3
Sources of Random Numbers

• Physical processes
• Noise in electrical signal generators, background
  electron scatter (atmospheric, CRT)
• Advantages natural, occurs in systems that use
  random numbers
• Disadvantage hard to measure and make into
  useful sequences, cannot specify distribution
•  Existing tables
• Last digits of phone number, log, sine, etc.
  tables
• Advantages already exist, many in digital form
• Disadvantages not really random, hard to use in
  simulations, cannot specify distribution

4
Sources of Random Numbers

•  Digital algorithms
• Last digits of arithmetic procedure
• Advantages can specify distribution, easy to use
  in computer simulation
• Disadvantages not really random (algorithm gt
  deterministic, will repeat sequence (cycles)
• Preferred method for computer based simulations
  long history, well developed
• Good algorithms have long periods and pass random
  tests widely used known as pseudo-random
  number generators

5
Definition

• Formally, common phrases "at random", "with equal
  chance", "equally likely" usually mean uniformly
  distributed
• The sequence 12345678901234567890,
  retrospectively consists of equally likely single
  digits not random since the next digit can be
  predicted from the previous successive digits
  depend on previous one gt not independent
• Therefore, random numbers are defined to be
  independent samples from a Uniform distribution
• Either real numbers on 0,1 or integers on 0,

6
Characteristics

• How do we know a sequence of numbers are
  uniformly distributed? satisfy tests that
  sequences known to be uniform satisfy
• Need statistical hypothesis testing
• In sequence of integers single digits, pairs,
  triplets, etc., have to be equally likely why?
• hard and laborious to check (e.g., for triplets
  would have to generate at least 1000 digits
  actually at least 5000 for Chi-squared test
  many more for longer subsequences) ultimately
  impractical

7
Narrow Definition

• Random integers can be generated from uniformly
  distributed random real numbers on 0,1
• Using Excel to generate integers uniformly
  distributed
• from 1 to N use trunc(rand()N1,0)
• from K to M use trunc(rand()(M1-K) K,0)
• Therefore, redefine random numbers to be
  independent real samples from 0,1

8
Linear Congruential Generator (LCG) for Uniform
Random Digits

• Preferred method begin with a seed, x0, and
  successively generate the next pseudo-random
  number by xi1 (axi c) mod m, for i
  0,1,2, where
• m is the largest prime less than largest integer
  computer can store
• a relatively prime to m
• c is arbitrary, often 0
• Let A be largest integer less than A, then N
  mod m N N/TT
• Accept LCG with m, a, and c which passes tests
  which are also passed by know uniform digits

9
Testing Random Numbers

• Most common are tests of distribution (in each
  case compare to expected by Chi-Square test - see
  Banks Section 7.4)
• Frequency tests Chi-Squared and
  Kolmogorow-Smirnov (count numbers in bins)
• Autocorrelation test compare autocorrelation at
  lags of 1, 2,
• Run test replace each number by 0 if lt.5 and 1
  is gt .5, count number of runs
• Gap test convert to integers, count number of
  intervening digits before same digit again
• Poker test convert to integers, take 5 digits,
  make poker hand out of it

10
Hypothesis Testing for Distributions

• H0 Sequence is uniformly distributed
• Test statistic c2 (pronounced chi-squared)
  applied to bins (single digits, pairs,
  triplets,etc.)
• p-value probability test statistic is as far
  from expected as it is also probability that of
  making a mistake by rejecting H0 if small
  (e.g., p lt .05), reject
• See spreadsheet RandomNum.xls

11
Empirical Random Variates

• When desired distribution does not have a know
  mathematical form and data is available, can
  generate variates from it
• Use inverse function technique
• Make table of values in increasing order with
  their associated cumulative probability
• Generate a random number (from uniform on 0,1
  interval) interpret this as a cumulative
  probability
• Find associated value
• Excel procedure vlookup() used previously

12
Non-Uniform Variate Generation

• Inverse function technique can be used to
  generate numbers from any statistical
  distribution calculate values from the
  distribution and use them as empirical values
  Banks Section 8.1
• Special techniques for particular distributions
  in rest of Banks Chapter 8