Games, Random Numbers and Introduction to simple statistics

1
Games, Random NumbersandIntroduction to simple
statistics
??? tsaiwn_at_csie.nctu.edu.tw

• PRNG
• Pseudo Random Number Generator

2
Agenda

• What is random number(??) ?
• How the random numbers generated ?
• rand( ) in C languages Linear Congruential
• Why call Pseudo random ? (P???)
• How to do true random ?
• Application of Rrandom number ?
• Other topics related to Random numbers
• Introduction to simple statistics (????)

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BATNUM game

• http//www.atariarchives.org/basicgames/showpage.p
  hp?page14
• An ancient game of two players
• One pile of match sticks (or stones)
• Takes turn to remove 1, maxTake
• (??? 1, ??? maxTake)
• ???????????? !
• Winning strategy ??

Games ??? Random Number! Why?
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Bulls and Cows Game

• http//5ko.free.fr/en/bk.html
• http//en.wikipedia.org/wiki/Bulls_and_cowsht
  tp//zh.wikipedia.org/zh-hant/E78C9CE695B0E
  5AD97http//boardgames.about.com/od/paperpencil
  /a/bulls_and_cows.htmhttp//pyva.net/eng/play/bk.
  html http//www.bullscows.com/index.phphttp//ww
  w.funmin.com/online-games/bulls-and-cows/index.php

Games ??? Random Number! Why?
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NIM Game

• http//en.wikipedia.org/wiki/Nim
• Nim is a two-player mathematical game of strategy
  in which players take turns removing objects from
  distinct heaps. On each turn, a player must
  remove at least one object, and may remove any
  number of objects provided they all come from the
  same heap.
• ???????????? !
• Winning strategy ??

Games ??? Random Number! Why?
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What is random number ?

• Sequence of independent random numbers with a
  specified distribution such as uniform
  distribution (equally probable)
• Actually, the sequence generated is not random,
  but it appears to be. Sequences generated in a
  deterministic way are usually called
  Pseudo-Random sequences.

?? http//www.gnu.org/software/gsl/manual/gsl-re
f_19.html
Normal distribution? exponential, gamma, Poisson,

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Turbo C ? rand( )?srand( )
Pseudo random number
ltstdlib.hgt

• define RAND_MAX 0x7fffu
• static unsigned long seed0
• int rand( )
• seed seed 1103515245 12345
• return seed (RAND_MAX1)
• void srand(int newseed)
• seed newseed

static global ?????KR??4.6?
??15? 1?binary
static ???file??function ???? seed
?? C ??? rand( ) ????? Normal Distribution!
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Unix ? gcc ? rand( )?srand( )
Pseudo random number
ltstdlib.hgt

• define RAND_MAX 0x7fffffffu
• static unsigned long seed0
• int rand( )
• seed seed 1103515245 12345
• return seed (RAND_MAX1)
• void srand(int newseed)
• seed newseed

static global ?????KR??4.6?
??31? 1?binary
Pseudo random number
?? Dev-Cpp ? gcc ???? 16 bits!
?? C ??? rand( ) ????? Normal Distribution!
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Random Number Generating Algorithms

• Linear Congruential Generators
• Simple way to generate pseudo-random numbers
• Easily cracked
• Produce finite sequences of numbers
• Each number is tied to the others
• Some sequences of numbers will not ever be
  generated
• Cryptographic random number generators
• Entropy sensors (i.e., extracted randomness)

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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 is relatively prime to m
• c is arbitrary
• 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

mod ?C/C/Java ?
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The use of random numbers
1. Simulation 2. Recreation (game programming) 3.
Sampling 4. Numerical analysis 5. Decision making
randomness an essential part of optimal
strategies ( in the game theory) 6. Game
program, . . .
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Uniform Distribution(???? )

• ? ????????

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Normal Distribution (???? )
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Standard Normal Distribution(??????)

• N(0, 1)
• ??? 0
• ??? 1

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????(the Normal Distribution)

• ?????,???????????????????????,?????????????????,?
  ???????????????????
• ? X ???????,?? X N(?,?2)?
• ???? ? ???,?2 ?????

?????????????(median)????(mode)????
?? C ??? rand( ) ????? Normal Distribution!
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Central Limit Theorem (CLT)(?????? )

• ??????? n ??,? n ??????????(independent and
  identically distributed, I.I.D.)?????(Random
  variable)????????????

????n?30?, ????????
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??? C ??????????

• include ltstdlib.hgt
• double randNormal( ) // ?????????
• int i
• double ans 0.0
• for(i1 ilt12 i) ans ans rand(
  )/(1.0RAND_MAX)return ans - 6.0 // N(0,
  1)

???? N(x, std2) ?
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Summary

• Pseudo-Random Number Generators(PRNG) depend
  solely on a seed, which determines the entire
  sequence of numbers returned.
• How to get true random ? ? change random seed
• How random is the seed?
• Process ID, UserID Bad Idea !
• Current time srand( time(0) ) // good
• If you use the time, maybe I can guess which
  seed you used (microsecond part might be
  difficult to guess, but is limited)

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Introduction to simple Statistics

• ???
• tsaiwn_at_csie.nctu.edu.tw

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???? ???

• ????????????96???????????????,????????????????????
  ?
• ???????????????,????????,?????????????????????????
  ?????????

??????? 100???????? "??" ?
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2010?????????

• ????????????????,?????????????????,?????50???,?
  ????32?????????????????????????????,???1960??
  ????
• ?????????,???????????,??????????????,????11?23?
  ??,????1273?,?95??????,??? 2.75?????

Sampling ??
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2005?????????
??????????????????
??????????????????,????????,???????1103???,?95???
???,?????2.95???
? ?????????
??????, ????????????
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2009????????
??????????/??????
?????????????????????,???????????????????????????
?????2009?11?10??11?????,?????932???????????,
??262?????????????????,???????3.2?????????????????
?????????????,???????????
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2008???? ????
??????????????????????????????? ???????20??????
???,??????,?????2008?1?12???????,1?13??16???6??1
0???,??1054?????,?95??????,?????3?
25
2006?10??????????
???????2006/9/27?9/28,????????????,????1112???????
????????????????????,?????2.9???
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2005??????????

• ??TVBS?11?21?22????????,?????????????????48,?????
  ????????27?????
• ??????????,??????????????????(BLOG)???,?????????
  ??,??2????,???????4?????
• ?????TVBS?????11?21??22??,?????1033?20????????,?95
  ?????,????????3.0?????

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1936 Presidential Election and Poll

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??1936???????

• ??????????????????????????????
• ??????????????
• ??????,?1929??1933?????????????
• ??????????????????The spender must go?
• ???????????????? (deficit financing)????Balance
  the budget of the American people first?
• ????????????????????
• ???Literary Digest????????57?43?????
• ?????????????????????
• ????1916??,????????????????
• ????????62?38?????????
• ?????-???-??
• ??Literary Digest??????????????,????????,???????56
  ?44?????
• ?????????????,??????????56?44?????

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Literary Digest???????

• ?????????????,????????,????????????????????
• ????????????????,????????
• ???????
• ????Literary Digest??????????????,???????????????
  
• ??????????
• ????????,?????????????,???20???,????????????(Land
  on),????????????????????

???????????
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Sample size vs. error of estimation

• When we use to construct a 95 confidence
  interval for ?, the bound on error of estimation
  is B
• n
• The estimated standard deviation of p is

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???????????

• 1-? Confidence Interval
• B the bound on error of estimation
• Using a conservative value of ? 0.5 in the
  formula for required sample size gives
• n ?(1-?) 0.5(1-0.5)
  1067.11
• Thus, n would need to be 1068 in order to
  estimate ? to within .03 with 95 confidence.

95??????,???????3???????
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Consider this program

• ?????????????????,??????????????
• ??????????
• n ?????????
• Average ????
• STD ? n ???????

?? ????????, ???????????????????, ??????????????!
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Descriptive Statistics

• Dispersion
• Range
• Standard deviation
• Variance
• N
• Not P (inferential stats)

• Distribution
• frequency distribution
• Histogram (???)
• Central tendency
• Mean
• Median (???)
• mode (??)

Dispersion ???????
Distribution ????? ??
Central tendency ???????
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Statistics

• Parameters (??????)
• Mean (???) - the average of the data
• Median (???) - the value of middle observation
• Mode (??) - the value with greatest frequency
• Standard Deviation (???) - measure of average
  deviation
• Variance (???) - the square of standard deviation
• Range (??) - ?? Max(B2B60) Min(B2B60)?

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Mean and Variance
Population Mean / Sample Mean
Sample Variance
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Standard Deviation

• Variance describes the spread (variation) of that
  data around the mean.
• Sample variance describes the variation of the
  estimates.
• Standard deviation s is the square root of s2

????? sqrt (???) ??????????
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Compute Variance without mean
Variance (??? ????/n) / n
From Wikipedia.org
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The Central Limit Theorem

• The probability distribution of sample means
  is a normal distribution
• If infinite number of samples with n gt 30
  observations are drawn from the same population
  where X ??(µ,s), then

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Central Limit Theorem (??????)

• For a population with a mean and a variance
  , the sampling distribution of the means of
  all possible samples of size n generated from the
  population will be approximately normally
  distributed - with the mean of the sampling
  distribution equal to and the variance equal
  to assuming that the sample size is
  sufficiently large.

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The Normal Distribution

• Described by
• (mean)
• (standard deviation ???)
• Variance ??? ??????
• Write as N( , ) ? N( , 2)
• Area under the curve is equal to 1
• Standard Normal Distribution

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Why is the Normal Distribution important?

• It can be a good mathematical model for some
  distributions of real data
• ACT Scores
• Repeated careful measurements of the same
  quantity
• It is a good approximation for different types of
  chance outcomes (like tossing a coin)
• It is very useful distribution to use to model
  roughly symmetric distributions
• Many statistical inference procedures are based
  on the normal distribution
• Sampling Distributions are roughly normal (TBC)

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Normal Distributions and the Standard Deviation
Normal Distribution Black line - Mean Red lines -
1 Std. Dev. from the mean (68.26
Interval) Green lines 2 Std. Dev. from the mean
(95.44 Interval) What about 3 Std. Dev. from
the mean?
95 Confidence interval 1.96 Std. Dev.
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68-95-99.7 Rule for Normal Curves
68.26 of the observations fall within ? of the
mean ?
95.44 of the observations fall within 2? of the
mean ?
99.74 of the observations fall within 3? of the
mean ?
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Notations

• It is important to distinguish between empirical
  and theoretical distributions
• Different notation for each distribution

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Density function of Normal Distribution

• The exact density curve for a particular normal
  distribution is described by giving its mean (?)
  and its standard deviation (?)
• density at x f(x)

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Confidence Intervals (CI) for µ,from a single
sample mean
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Confidence Interval? (1/2)

• ???????????????,??????(random number)?????,???????
  ??????X1,????X2,???n??,??X1?X2...Xn?n???,?n???????
  ,?????????? ???,????n????????,?????????????
• ????????????????????(sample mean)??
• ????????????Confidence Interval
  (????)?????(significance level)?

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Confidence Interval? (2/2)

• ????,?????????????????????????????,????????
• ??????????????????(probability bound)?????????????
  ????c1????????c2,??????????1 a
  ,????????????????µ(sample mean)???c1?c2??????
• Probability c1 lt µ lt c2 1 a

???(c1, c2)??????????(confidence
interval) a??????(significance
level) 100(1-a)??????(confidence
level),?????? 1-a??????(confidence coefficient)?
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??????????????????

• ????16??????393??????????????,
• ???1033???????
• ????????,???????????????????,?????????????????,???
  ??16??????????????(????)???
• ?????????,?????????????????

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Hypothesis Testing?????

• The null hypothesis for the test is that all
  population means (level means) are the same.
  (H0)
• The alternative hypothesis is that one or more
  population means differ from the others. (H1)

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PRNG ????

• ?? http//gogle.com ? PRNG ??
• ANSI X9.17 PRNG
• (PRNG Pseudo Random Number Generator)
• Von Neumann ??? middle square method
• Von Neumann architecture ?
• PRNG in RC4 (RC4?? 802.11 ???????)
• http//www.rsa.com
• http//www.wisdom.weizmann.ac.il/itsik/RC4/rc4.ht
  ml
• WEP RC4 Stream cipher

52
ANSI X9.17 PRNG

• Use 3DES and a key K
• Ti Ek(current timestamp)
• outputi Ek(Ti ? seedi)
• seedi1 Ek(Ti ? outputi)
• Weaknesses
• Only 64 bits are used for Ti
• seedi1 can be easily predicted if state
  compromise

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Jon von Neumann 1946 suggested the production of
random number using arithmetic operations of a
computer, "middle square", square a
previousrandom number and extract the middle
digits, Example generate 10-digit numbers,
was 5772156649, square 33317792380594909201t
he next number is 7923805949
Middle square
"middle square" has proved to be a comparatively
poor source of random numbers. If zero appear as
a number of the sequence, it will continually
perpetuate itself.
54
Von Neumann architecture (http//wikipedia.org/)

• The term von Neumann architecture refers to a
  computer design model that uses a single storage
  structure to hold both programs and data. The
  term von Neumann machine can be used to describe
  such a computer, but that term has other meanings
  as well. The separation of storage from the
  processing unit is implicit in the von Neumann
  architecture.
• The term "stored-program computer" is generally
  used to mean a computer of this design.

Von Neumann bottle neck ?
55
Seeding RC4
RC4 PRNG (1/2)

• for(I 0 I lt 256 I)
• SI I
• for (I J 0 I lt 256 I)
• j SI KI klen
• SWAP(SI, SJ)
• I J 0

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RC4 PRNG (2/2)

• rc4byte()
• I
• J SI
• SWAP(SI, SJ)
• return (S SI SJ )

Byte version
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WEP RC4 ??? (http//rsa.com)
Random bit stream b
Plaintext bit stream p
Ciphertext bit stream c
?
XOR
Decryption works in the same way p c ? b
WEP Wired Equivalent Privacy
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Games, Random NumbersandIntroduction to simple
statistics

• ????
• http//www.csie.nctu.edu.tw/tsaiwn/introcs/
• http//gogle.com/