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Games, Random Numbers and Introduction to simple statistics. PRNG Pseudo R andom N umber G enerator. 蔡文能 [email protected] Agenda. What is random number ( 亂數 ) ? How the random numbers generated ? rand( ) in C languages: Linear Congruential

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Games, Random NumbersandIntroduction to simple statistics

PRNG

PseudoRandom Number Generator

蔡文能

[email protected]


Agenda

  • What is random number(亂數) ?

  • How the random numbers generated ?

    • rand( ) in C languages: Linear Congruential

  • Why call “Pseudo random” ? (P不發音)

  • How to do “true random” ?

  • Application of Rrandom number ?

  • Other topics related to Random numbers

  • Introduction to simple statistics (統計簡介)


BATNUM game

  • http://www.atariarchives.org/basicgames/showpage.php?page=14

  • An ancient game of two players

  • One pile of match sticks (or stones)

  • Takes turn to remove [1, maxTake]

  • (至少拿 1, 至多拿 maxTake)

  • 可規定拿到最後一個贏或輸 !

  • Winning strategy ??

Games 須用到 Random Number! Why?


Bulls and Cows Game

http://5ko.free.fr/en/bk.html

http://en.wikipedia.org/wiki/Bulls_and_cowshttp://zh.wikipedia.org/zh-hant/%E7%8C%9C%E6%95%B0%E5%AD%97http://boardgames.about.com/od/paperpencil/a/bulls_and_cows.htmhttp://pyva.net/eng/play/bk.htmlhttp://www.bullscows.com/index.phphttp://www.funmin.com/online-games/bulls-and-cows/index.php

Games須用到 Random Number! Why?


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.

  • 可規定拿到最後一個贏或輸 !

  • Winning strategy ??

Games 須用到 Random Number! Why?


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.

參考 http://www.gnu.org/software/gsl/manual/gsl-ref_19.html

Normal distribution? exponential, gamma, Poisson, …


Pseudo random number

<stdlib.h>

Turbo C++ 的 rand( )與srand( )

static global 變數請參考K&R課本4.6節

就是15個 1的binary

#define RAND_MAX 0x7fffu

static unsigned long seed=0;

int rand( ) {

seed = seed * 1103515245 + 12345;

return seed % (RAND_MAX+1);

}

void srand(int newseed) {

seed = newseed;

}

static使其它file裡的function 看不見這 seed

注意 C 語言的 rand( )生出的不是 Normal Distribution!


Pseudo random number

<stdlib.h>

Unix 上 gcc 的 rand( )與srand( )

static global 變數請參考K&R課本4.6節

#define RAND_MAX 0x7fffffffu

static unsigned long seed=0;

int rand( ) {

seed = seed * 1103515245 + 12345;

return seed % (RAND_MAX+1);

}

void srand(int newseed) {

seed = newseed;

}

就是31個 1的binary

Pseudo random number

注意 Dev-Cpp 的 gcc 亂數只有 16 bits!

注意 C 語言的 rand( )生出的不是 Normal Distribution!


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)


Linear Congruential Generator (LCG) for Uniform Random Digits

  • Preferred method: begin with a seed, x0, and successively generate the next pseudo-random number by xi+1 = (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 (就是只取整數部份),

    then N mod m = N – [N/T]*T

  • Accept LCG with m, a, and c which passes tests which are also passed by know uniform digits

mod在C/C++/Java 用 %


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, . . .


Uniform Distribution(齊一分配 )

  • 在 發生的機率皆相同


Normal Distribution (常態分配 )


Standard Normal Distribution(標準常態分配)

  • N(0, 1)

  • 平均是 0

  • 標準差 1


常態分配(the Normal Distribution)

  • 在統計學中,最常被用到的連續分配就是常態分配。在真實世界中,常態分配常被用來描述各種變數的行為,如考試成績、體重、智商、和商店營業額等。

  • 若 X 為常態隨機變數,寫成 X ~ N(,2)。

    • 其中參數  為均數,2為變異數。

常態隨機變數的均數、中位數(median)、與眾數(mode)均相同。

注意 C 語言的 rand( )生出的不是 Normal Distribution!


Central Limit Theorem (CLT)(中央極限定理 )

  • 如果觀察值數目 n 增加,則 n 個獨立且具有相同分配(independent and identically distributed, I.I.D.)的隨機變數(Random variable)之平均數向常態分配收斂。

樣本大小n≧30時, 趨近於常態分配。


如何用 C 生出常態分配的亂數?

#include <stdlib.h>

double randNormal( ) { // 標準常態分配產生器

int i;

double ans =0.0;

for(i=1; i<=12; ++i) ans = ans + rand( )/(1.0+RAND_MAX);return ans - 6.0; // N(0, 1)

}

如何生出 N(x, std2) ?


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)


Introduction to simple Statistics

蔡文能

[email protected]


大考中心 …教育部…

  • 大學入學考試中心指出民國96年指考國文科較接近「常態分布」,即中等程度的人數最多、高分、低分人數較少。

  • 教育部修訂資賦優異學生鑑定標準,自九十六學年度起,各類資優鑑定標準已提高為「平均數正二個標準差或百分等級九十七以上」。

請問照這樣標準 100人中大約有幾人是 "資優" ?


2010大高雄市長選舉民調

  • 目前將在明年登場的大高雄市長選舉,根據《財訊》雙週刊所公佈的最新民調顯示,高雄市民有50%挺陳菊,朱立倫僅有32%支持度;若是由國民黨內佈局明年市長最明顯的立委黃昭順對上陳菊,則更有19%:60%的大段差距。

  • 本次《財訊》雙週刊民調,係委託山水民意研究公司,以北、高兩市住宅電話隨機取樣,高雄市於11月2~3日進行,有效樣本1273人,在95%的信心水準下,誤差約 ±2.75個百分點。

Sampling 抽樣


2005南投縣長選舉大調查

請問您南投縣最急需改善的問題是什麼?

中時電子報是於十一月八、九、十日三天,利用電話隨機抽樣,成功訪問南投縣1103名民眾,在95%的信心水準下,正負誤差為2.95%以下。

註: 結果是李朝卿當選。

年底縣長選舉, 請問您支持哪一位參選人?


2009南投縣長選舉民調

聯合報系民意調查中心/電話調查報導

國民黨李朝卿聲勢較半個月前上揚十八個百分點,目前以四成八的支持率大幅領先民進黨李文忠的百分之三十。

這次調查於2009年11月10日至11日晚間進行,成功訪問了932位設籍南投縣的成年選民,另有262人拒訪。在百分之九十五的信心水準下,抽樣誤差在正負3.2個百分點以內。調查是以南投縣住宅電話為母體作尾數兩位隨機抽樣,調查經費來自聯合報社。


2008總統大選 蘋果民調

這是台灣《蘋果日報》委託中山大學社科院民意調查研究中心所做最新民調;

此民調針對全台20歲以上有投票權公民,進行電話訪問,調查時間為2008年1月12日“立委”選舉隔天,1月13日至16日晚上6時至10時之間,共有1054個成功樣本,在95%信心水準之下,正負誤差約3%。


2006年10月台北市長候選人民調

中時電子報是於2006/9/27到9/28,以中華民國家戶電話為樣本,成功訪問1112名居住地為台北市的受訪者。在百分之九十五的信心水準之下,正負誤差為2.9%以下。


2005台北縣長選舉民意調查

  • 根據TVBS在11月21至22日的民意調查顯示,國民黨台北縣候選人周錫瑋的支持度為48%,民進黨的候選人羅文嘉則獲得27%的支持度。

  • 此次民調和上月前相比,繼永洲案爆發後及日前沸騰的“瑋哥部落格(BLOG)”的抹黑,周錫瑋的支持度不降反升,多了2個百分點,羅文嘉則是下降4個百分點。

  • 這份民調是TVBS民調中心在11月21日到22日間,成功訪問了1033位20歲以上的台北縣民,在95%信心水準下,抽樣誤差約為正負3.0個百分點。


1936 Presidential Election and Poll


背景:1936年美國總統選舉

  • 法蘭克羅斯福總統爭取連任、肯薩斯州州長蘭登為共和黨總統候選人

  • 美國經濟正由大蕭條中逐漸恢復

    • 九百萬人失業,於1929年至1933年間實際所得降低三分之一。

    • 蘭登州長選戰主軸為「小政府」。口號為The spender must go。

    • 羅斯福總統選戰主軸為「擴大內需」 (deficit financing)。口號為Balance the budget of the American people first。

  • 宣稱一:大部分的觀察家認為羅斯福總統將大勝

  • 宣稱二:Literary Digest雜誌認為蘭登將以57%對43%贏此選戰。

    • 此數字乃根據於二百四十萬人之民意調查結果。

    • 該機構自1916年起,皆能依照其預測辦法作正確的預測。

  • 選舉結果:羅斯福以62%對38%贏此選戰。為什麼?

  • 新興競爭者-蓋洛普-民調:

    • 依據Literary Digest雜誌所取的二百四十萬人樣本中,蓋洛普抽樣三千人,而預測蘭登將以56%對44%贏此選戰。

    • 依據自己所取的五萬人樣本中,蓋洛普預測羅斯福將以56%對44%贏此選戰。


Literary Digest雜誌錯在那裡?

  • 取樣辦法:郵寄一千萬份的問卷,回收二百四十萬份,但問卷對象係從電話簿及俱樂部會員中選取。

    • 在當時僅有一千一百萬具住宅用電話,但九百萬人失業。

      可能問題的所在:

  • 取樣偏差:Literary Digest雜誌的取樣中包含過多的有錢人,而該年貧富間選舉傾向相距極大。

  • 拒回答偏差:低回收率。

    • 以芝加哥一地為例,問卷寄給三分之一的登記選民,回收約20%的問卷,其中超過一半宣稱將選蘭登(Landon),但選舉結果卻是羅斯福拿到三分之二的選票。

抽樣的樣本要多少才夠?


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


抽樣的樣本要多少才夠?

  • 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%信心水準之下,抽樣誤差在正負3個百分點以內。


Consider this program

  • 台北車站廣場打算設置一台體重統計機,任何人站上去後立刻顯示其體重

  • 並且立即顯示以下統計:

    n : 共已多少人在此量過

    Average : 平均體重

    STD : 這 n人的體重標準差

注意:因為可能會很多人, 所以不能把所有量過的體重都記在記憶體內, 機器也沒有硬碟或其他儲存裝置!


Distribution

frequency distribution

Histogram (長條圖)

Central tendency

Mean

Median (中位數)

mode (眾數)

Dispersion

Range

Standard deviation

Variance

N

Not P (inferential stats)

Descriptive Statistics

Dispersion 資料之散亂;發散

Distribution 資料之分佈; 分配

Central tendency 資料之集中趨勢


Statistics

  • Parameters (常見統計參數)

    • Mean (平均數) ─ the average of the data

    • Median (中位數)─ the value of middle observation

    • Mode (眾數) ─ the value with greatest frequency

    • Standard Deviation (標準差) ─ measure of average deviation

    • Variance (變異數) ─ the square of standard deviation

    • Range (範圍) ─ 例如 Max(B2:B60) ~ Min(B2:B60)?


若是全部資料而不是抽樣, 則除以 n而不是除以 n -1, 此即 population variance

Mean and Variance

Population Mean / Sample Mean

Sample Variance


Standard Deviation

  • Variance describes the spread (variation) of that data around the mean.

  • Sample variance describes the variation of the estimates.

  • Standard deviations is the square root of s2

標準差就是sqrt (變異數); 阿就是變異數的平方根


Compute Variance without mean

Variance = (平方和 – 和的平方/n) / n

From Wikipedia.org


The Central Limit Theorem

  • The probability distribution of sample means is a normal distribution

  • If infinite number of samples with n > =30 observations are drawn from the same population where X ~ ??(μ,σ), then


Central Limit Theorem (中央極限定理)

  • For a population with a mean and a variance , the sampling distribution of the means of all possible samples of size ngenerated 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.


The Normal Distribution

  • Described by

    • (mean)

    • (standard deviation; 標準差)

    • Variance 變異數 = 標準差的平方

  • Write as N( , ) 或 N( , 2)

  • Area under the curve is equal to 1

  • Standard Normal Distribution


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…)


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.


68.26%

95.44%

-

µ

+

-2

µ

+2

99.74%

-3

+3

µ

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 


Notations

  • It is important to distinguish between empirical and theoretical distributions

  • Different notation for each distribution


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) =


Confidence Intervals (CI) for µ,from a single sample mean


Confidence Interval? (1/2)

  • 當我們使用軟體去模擬真實環境時,通常會用亂數(random number)模擬很多次,假設第一次模擬的結果數據是X1,第二次是X2,重覆了n次後,就有X1、X2...Xn共n個數據,這n個數據不盡相同,到底那個才是正確的? 直覺上,把得到的n個結果加總求平均,所得到的值應該比較能相信。

  • 但是我們可以有多少程度的去相信這個平均值(sample mean)呢?

  • 這個問題討論的就是所謂的Confidence Interval (信賴區間)與顯著水準(significance level)。


Confidence Interval? (2/2)

  • 在實務上,想要在有限個模擬數據結果中得到一個較完美接近真實結果的數據,其實是不可能的。

  • 因此我們能做的就是去求得一個機率範圍(probability bound)。若我們可以得到一個機率範圍的上限c1和一個範圍的下限c2,則就有一個很高的機率1 – α ,會使得每次所得到的模擬結果平均值μ(sample mean)都落在c1到c2的範圍之間。

    Probability { c1 <= μ <= c2} = 1 –α

我們把(c1, c2)這個範圍稱為信賴區間(confidence interval);

α稱為顯著水準(significance level);

100(1-α)%稱為信心水準(confidence level),用百分比表示;

1-α稱為信心係數(confidence coefficient)。


為何簡單隨機抽樣是個合理的抽樣方法?

  • 試想抽取16所醫院來預測393所醫院的平均出院病人數的例子,

    • 共有約1033種的不同樣本。

    • 依據中央極限定理,所得到的平均出院病人數分佈像個鐘形曲線,其中心位於所有醫院的平均出院病人數,且大多數的16所醫院平均出院病人數都離中心(大數法則)不遠。

      較有保障的抽樣辦法,被選取的樣本應使用隨機的原理取得。


Hypothesis Testing假設之檢定

  • 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)


PRNG 相關補充

  • 請用 http://gogle.com打 “PRNG” 查看

  • ANSI X9.17 PRNG

    (PRNG = Pseudo Random Number Generator)

  • Von Neumann想出的 middle square method

  • Von Neumann architecture ?

  • PRNG in RC4 (RC4用於 802.11 無線網路加解密)

    • http://www.rsa.com

    • http://www.wisdom.weizmann.ac.il/~itsik/RC4/rc4.html

  • WEP : RC4 Stream cipher


ANSI X9.17 PRNG

  • Use 3DES and a key K

  • Ti = Ek(current timestamp)

  • output[i] = Ek(Ti seed[i])

  • seed[i+1] = Ek(Ti output[i])

  • Weaknesses

    • Only 64 bits are used for Ti

    • seed[i+1] can be easily predicted if state compromise


Middle square

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 33317792380594909201the next number is 7923805949

"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.


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 ?


Seeding RC4

RC4 PRNG (1/2)

for(I = 0; I < 256; I++)

S[I] = I;

for (I = J = 0; I < 256; I++) {

j += S[I] + K[I % klen];

SWAP(S[I], S[J]);

}

I = J = 0;


RC4 PRNG (2/2)

rc4byte()

{

I++;

J += S[I];

SWAP(S[I], S[J]);

return (S[ S[I] + S[J] ]);

}

Byte version


Pseudo-random number generator

Encryption Key K

WEP: RC4 加解密 (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


Games, Random NumbersandIntroduction to simple statistics

謝謝捧場

http://www.csie.nctu.edu.tw/~tsaiwn/introcs/

http://gogle.com/


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