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Transcript of M03-13 Muhammad Khan Muneer (Presentation on Random Numbers)
7/27/2019 M03-13 Muhammad Khan Muneer (Presentation on Random Numbers)
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By
Jarry Jafery (M15-13)
Muhammad Khan Muneer (M03-13)
7/27/2019 M03-13 Muhammad Khan Muneer (Presentation on Random Numbers)
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Lacking a definite plan, purpose, or pattern
A set where each of the elements has equalprobability of occurrence
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A sequence in which each term isunpredictable
D. H. Lehmer (1951)When discussing a sequence of random numberseach number drawn must be statisticallyindependent of the others.
Examples between 1 and 100 29, 95, 11, 60, 22
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The Chinese were perhaps the earliest people toformalize odds and chance 3,000 years ago.
The Greek philosophers discussed randomness atlength, but only in non-quantitative forms.
It was only in the sixteenth century that Italianmathematicians began to formalize the odds
associated with various games of chance.
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In 1937 Kendall and smith obtained 100,000 random
numbers grouped in two’s and four’s from machine. Also
F isher and Yates generated 15,000 random numbers
arranged in two’s which were taken from A.J Thompson’s table of logarithm.
In 1951, Derr ick Hennery Lehmer invented the linear
congruential generator, used in most pseudorandomnumber generator today.
In 1955 Rand Corporation built a special machine to
generate pseudorandom binary bits of 0 and1 that werethen used to produce a table of one million random
decimal digits.
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John Von Neumann was a pioneer in
computer-based random numbergenerators.
With the spread of the use of computersalgorithmic pseudorandom number
generators replaced random number tables,
and “true” random number generators
(Hardware random number generators) are
only used in a few cases.
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Almost all network security protocols rely on the
randomness of certain parameters
Nonce - used to avoid replay
Session Key
Unique parameters in digital signatures
Monte Carlo Simulations -
is a mathematical technique for numerically solving differential
equations. Randomly generates scenarios for collecting statistics.
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Simulation
Computer ProgrammingDecision Making
Recreation
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Simulate natural phenomena on a computer
Used for experiments in sterile conditions tomake them more realistic
Useful in all of the Applied Disciplines
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Test program effectiveness
Test algorithm correctness
Instead of all possible inputs use a few random
numbers Microsoft has used this logic in testing their software
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When an “unbiased” decision is needed
Fixed decision can cause some algorithms to runmore slowly
Good way of choosing who goes first Sporting events
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Lottery Equal odds
The KY Lottery uses Microsoft Excel’s RNG for“various second chance drawings“
Casinos Provides a chance for “luck”
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Video Games Random events keep games entertaining
Q-bert
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True random numbersPseudo-random numbers
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Truly random - is defined as exhibiting ``true''randomness, such as the time between ``tics''from a Geiger counter exposed to a radioactiveelement
Pseudorandom - is defined as having theappearance of randomness, but neverthelessexhibiting a specific, repeatable pattern.
numbers calculated by a computer through a
deterministic process, cannot, by definition, berandom
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Examples of random numbers:
Noise in electrical circuits
Radioactive decay Flow of water from a vessel
Session keys
Numbers to be hashed with passwords
Prepaid card numbers
Nonce
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Anyone who considers arithmetical methods ofproducing random digits is, of course, in a state of sin.
John Von Neumann (1951)
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Properties of pseudo-random numbers
• Continuous numbers between 0 and 1
• Probability of selecting a number in interval (a,b)~ (b-a) – i.e. Uniformly distributed
• Numbers are statistically independent
• Can’t really generate random numbers
• Also, want fast and repeatable
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Muhammad Khan Muneer
M03-13
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How to generate random numbers
Table look-up
Computer generation: these values cannot betruly random and a computer cannot express anumber to an infinite number of decimal places.
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Measurements of electric current passing through awire at some time point.
Inflow and outflow of water at a reservoir at sometime point.
Timing of keystrokes when a user enters a password.
Measurement of air turbulence due to the movementof hard drive heads.
Precise measurement of current leakage from a CPU
or any other system component. Measurement of timing skew between two systems’
timers: A hardware timer
A software timer
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Selection of random numbers from random numbertables:
A practical method of selecting a random sample is tochoose units one-by-one with the help of a Table ofrandom numbers. By considering two-digits numbers,we can obtain numbers from 00 to 99, all havingthe same frequency. Similarly, three or moredigit numbers may be obtained by combining three ormore rows or columns of these Tables. The simplestway of selecting a sample of the required size
is by selecting a random number from 1 to N andthen taking the unit bearing that number. Thisprocedure involves a number of rejections since allnumbers greater than N appearing in the Tableare not considered for selection.
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So, the selected numbers are, therefore,modified and some of the modificationprocedures are:
Remainder ApproachQuotient Approach
Independent Choice of Digits
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Let N be an r-digit number and let its r-digithighest multiple be N'. A random number k is chosen from 1 to N' and the unit with theserial number equal to the remainderobtained on dividing k by N, is selected. Ifthe remainder is zero, the last unit isselected.
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Example:
Let N = 127, the highest three-digit multiple of127 is 889. For selecting a unit, one random
number from 001 to 889 has to be selected.Let the random number selected be 503.Dividing 503 by 127, the remainder is122. Hence, the unit with serial number 122
is selected in the sample.
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Let N be an r-digit number and let its r-digithighest multiple be N' such that N' / N = d. Arandom number k is chosen from 0 to (N'-1).Dividing k by d, the quotient q is obtainedand the unit bearing the serial number(q - 1) is selected in the sample.
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Let N = 122 and hence N' = 976 and d = 976 /122 = 8. Let the three-digit random numberchosen be 503 which lies between 0 and975. Dividing 503 by 8, the quotient is 62and hence the unit bearing serial number(62-1) = 61 is selected in the sample.
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This method, suggested by Mathai (1954),consists of the selection of two randomnumbers which are combined to form onerandom number. One random number ischosen according to the first digit and otheraccording to the remaining digits of thepopulation size. If the number chosen is 0,the last unit is chosen. But if the numbermade up is greater than or equal to N, thenumber is rejected and the operation isrepeated.
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Example:
Select a random sample of 11 households froma list of 112 households in a village by usingthe 3-digit random numbers given in columns 1
to 3, 4 to 6 and so on of the random numbertable and rejecting numbers greater than 112(also the number 000), we have for the samplebearing serial numbers 033, 051, 052, 099, 102,081, 092, 013, 017, 076 and 079. In the
above procedure, a large number ofrandom numbers is rejected. Hence, acommonly used device i.e., remainderapproach, is employed to avoid the rejection ofsuch large numbers.
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The greatest three-digit multiple of 112 is 896.By using three digit random numbers as above,the sample will comprise of households withserial numbers 086, 033, 049, 097, 051, 052,
066, 107, 015, 106 and 020. In case the quotientapproach is applied, the 3-digit multiple of 112is 896 and 896/112 = 8. Using the same randomnumber and dividing them by 8, we have thesample of households with list numbers 025,
004, 020, 026, 006, 006, 092, 041, 085, 027 and086 with the replacement method and with listnumbers 025, 004, 020, 026, 006, 092, 041, 085,027, 086 and 042 without the replacementmethod.
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There are many methods and algorithms havebeen developed for generation of randomnumbers. The most common ones are thefollowing:
Midsquare Method
Linear Congruential Generator (LCG)
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Random Number Seed:
Virtually all computer methods of randomnumber generation start with an initial
random number seed. This seed is used togenerate the next random number and thenis transformed into a new seed value.
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Midsquare method consists of following steps:
1. Start with an initial seed (e.g. a 4-digitinteger).
2. Square the number.
3. Take the middle 4 digits.
4. This value becomes the new seed. Dividethe number by 10,000. This becomes therandom number. Go to step 2.
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Example:
x 0 = 5497
x 1: 54972 = 30217009 x 1 = 2170, R1 = 0.2170
x 2: 21702
= 04708900 x 2 = 7089, R2 = 0.7089 x 3: 70892 = 50253921 x 3 = 2539, R3 = 0.2539
Drawback: It’s hard to state conditions for picking initial
seed that will generate a “good” sequence.
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Example:
“Bad” sequences:
x 0
= 5197 x 1: 51972 = 27008809 x 1 = 0088, R1 = 0.0088 x 2: 00882 = 00007744 x 2 = 0077, R2 = 0.0077 x 3: 00772 = 00005929 x 3 = 0059, R3 = 0.0059
x i = 6500
x i+1: 65002=42250000 x i+1=2500, Ri+1= 0.0088 x i+2: 25002=06250000 x i+2=2500, Ri+1= 0.0088
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Proposed by Lehmer in 1951 Let Zi be the ith number (integer) in the sequence
Zi = (aZi-1+c)mod(m)
Zi
{0,1,2,…,m-1}
Where Z0 = initial seed
a = multiplier
c = incrementm = modulus
DefineRi = Zi /m (to obtain U[0,1) value)
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Start with random seed Z0 Where Z0 < m = largest possible integer on machine
Recursively generate integers between 0 and M
Zi = (a Zi-1 + c) mod m
Use R = Z/m to get pseudo-random number(avoid 0 and 1)
When c = 0 Called Multiplicative CongruentialGenerator
When c > 0 Mixed LCG
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Example:
16-bit machine
a = 1217
c = 0Z0 = 23
m = 215-1 = 32767
Z1 = (1217*23) mod 32767 = 27991
U1 = 27991/32767 = 0.85424
Z2 = (1217*27991) mod 32767 = 20134
U2 = 20134/32767 = 0.61446
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What makes one LCG better than
another?
A full period (full cycle) LCGgenerates all m values before it cycles.
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The period of an LCG is m (full period or fullcycle) if and only if
If q is prime that divides m, then q divides a-1c and m should be relatively prime, i.e., g.c.d.
= 1. (The only positive integer that divides both mand c is 1)
If 4 divides m, then 4 divides a-1. (e.g., a = 1,5, 9, 13,…)
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m = 2B where B is the no. of bits in the machineis often a good choice to maximize the period.
If c = 0, we have a power residue ormultiplicative generator.
When c > 0 Mixed LCG Note that Zn = (aZn-1) mod(m) Zn = (anZ0)
mod(m).
If m = 2B, where B is the no. of bits in the
machine, the longest period is m/4 (best one can do) if and only if Z0 is odd
a = 8k + 3, k Z+ (5,11,13,19,21,27,…)
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Never “invent” your own LCG. It will probably
not be “good.”
All simulation languages and many software
packages have their own PRN generator. Most usesome variation of a linear congruential generator.
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Theoretical tests
Prove sample moments over entire cycle arecorrect
Lattice (plotted in 2 or 3 dimensions) structure ofLCGs “random numbers fall mainly in the planes”
(Marsaglia)
Spacing hyperplanes: the smaller, the better
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Empirical tests Uniformity
Compute sample moments
Goodness of fit
Independence Gap Test
Runs Test
Poker Test
Autocorrelation Test
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This test consists of following steps:
Divide “n” observations into “k ” equal intervals.
Do a frequency count f i, i=1,2,…,k
Compute
2
2 1
k
i
i
n f
k n
k
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2
2
1
k i i
i i
f np
np
1 1, 2, 3, ,iwhere p and i k
k
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Data Classification
ei = expected number of observations ininterval i = n pi = n / k, i = 1, 2, …, k
• • •
f 1 f 2 f k-1 f k
01
k
2
k
k
2
k
k
1
k
e1 e2 ek-1 ek
1
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Repeat test “m” times with independent
samples of size “n”.
Do Not
Reject HOReject HO
k 1,2
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Important things to note here are:
Choose the intervals evenly
Choose the intervals such that you would expect
each class to contain at least 5 or 10 observations pi should (ideally) be small (<.05)
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Example
n = 1000
k = 10
pi = 1/k = 0.1 ei = npi = 100
2
2 1 6.28
k
i i
i
i
f np
np
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i Intervals i f i i f np
2
i i f np
2
i i
i
f np
np
1 [0, .1) 87 -13 169 1.69
2 [.1, .2) 93 -7 49 0.49
3 [.2, .3) 113 13 169 1.694 [.3, .4) 106 6 36 0.36
5 [.4, .5) 108 8 64 0.64
6 [.5, .6) 99 -1 1 0.01
7 [.6, .7) 91 -9 81 0.818 [.7, .8) 95 -5 25 0.25
9 [.8, .9) 103 3 9 0.09
10 [.9, 1.] 105 5 25 0.25
1000 6.28
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2
2 1 6.28
k
i i
i
i
f np
np
2
0.05 (9) 16.919
Do not reject H0 : U(0,1)
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Runs of increasing and decreasing numbers
This test consists of following steps:
Assign + if xi < xi+1, assign - if xi>xi+1
Test Statistic: S = number of runs up AND down(sequence of + and -)
E(S) = (2N-1)/3
Var(S) = (16N-29)/90
Use Normal approximation for N>30
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Example:
N = 15
S = 8 ~ N(µ = 29/3, 2 = 211/90)
Maximum value for S:N-1; negative dependency
Minimum value for S: 1; positive dependency
0.87 0.15 0.23 0.45 0.69 0.32 0.3 0.19 0.24 0.18 0.65 0.82 0.93 0.22 0.81
- + + + - - - + - + + + - +
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Normal Curve Rejection Regions
REJECT (-ve)REJECT
Do Not
REJECT
H0 : Independence
HA : Dependence
Z/2-Z
/2
Reject H0 in favor of HA if
Z = (S - (2N-1)/3) / (16N-29/90)1/2 Z/2 or Z Z
/2
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