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• Properties of Random Numbers
• Generating Random Numbers
• Testing Random Numbers
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• Key Properties
– Uniformity
– Independence
• Density function (continuous!) f(x) = {1 for 0 < x < 1, 0 otherwise
• Moments
E(R) = 1/2 V(R) = 1/12
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• Random Numbers vs Pseudo-random
Numbers
• Requirements of a RNG routine
– Speed
– Portable
– Long Cycle
– Replicable RN
– Uniform and Independent RN’s
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• Linear Congruential Method
X i+1
= (a X i
+ c) mod m
R i
= X i
/m
• Note: Only values from the set
I = {0,1/m,2/m,…,(m-1)/m} are obtained
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Random Number Generation -contd
• Longest Possible Period (P)
– If m = 2 b and |c| > 0 , P = m
– If m = 2 b and c = 0 , P = m/4
– If m = prime and c = 0 , P = m-1
• Example:
X i+1
= (7 5 X i
) mod (2 31 -1)
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Random Number Generation -contd
• Combined Congruential Generators. Two distinct congruential generators can be combined to obtain PRN’s with longer periods.
X i+1
= (

(-1) j-1 X i,j
) mod (m
1
- 1)
R i
= X i
/m
1
, X i
> 0 ; R i
= (m
1
-1)/m
1
, X i
> 0
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Testing Random Numbers
• Null Hypotheses
H
0
: R
• Tests i
~ U[0,1] ; H
0
: R i
~ independent
– Frequency test
–
Runs test
–
Autocorrelation test
–
Gap test
– Poker test
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Kolmogorov-Smirnov Frequency Test
1.- Arrange data in increasing value
2.- Compute D + , D and D
3.- Find critical D c
(Handout) for given a
4.- Accept or reject the null hypothesis.
5.- Example: Stat::Fit
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Chi-Square Frequency Test
• The Chi static compares observed frequencies of occurrence of PRN’s in selected subdomains against expected frequencies derived from the U distribution function. See Stat::Fit
X
0
2 =
 n
(O i
- E i
) 2 /E i
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• Run: sequence of similar events
• Runs up and runs down (independence)
– Maximum number of runs (N numbers) = N-1
– mean = (2N-1)/3; variance = (16N-29)/90
– Test hypothesis against normal distribution.
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• Runs above and below the mean
– Maximum number of runs (N numbers, n1 above and n2 below the mean) = n1+n2
– mean = 2 n1 n2/N + 1/2
– variance = 2 n1 n2 (2 n1 n2 - N)/N 2 (N-1)
– Test hypothesis against normal distribution.
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• Runs length
– Test hypothesis against Chi square distribution
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• Seek the autocorrelation between every m numbers (I.e. dependence)
• Null Hypothesis H
0
:
 im
= 0
• Note: If values are uncorrelated,  im has normal distribution. So, test hypothesis against normal distribution.
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• Gap: Interval of recurrence of same digit.
• Monitor Frequency of gaps and test
1.- Specify the cdf F(x) = 1-0.9 x+1
2.- Arrange the observed gaps into S(x)
3.- Find D and Dc
4.- Accept or reject the null hypothesis.
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• Frequency of repetition of certain digits in a series
• Null hypothesis is tested againts the Chisquare distribution.
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