Games, Random Numbers and Introduction to simple statistics

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1 Games, Random Numbers and Introduction to simple statistics
蔡文能 PRNG Pseudo Random Number Generator 蔡文能 C/C++ 程式設計

2 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) 蔡文能 C/C++ 程式設計

3 Games 須用到 Random Number! Why?
BATNUM game 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? 蔡文能 C/C++ 程式設計

4 Games 須用到 Random Number! Why?
Bulls and Cows Game Games 須用到 Random Number! Why? 蔡文能 C/C++ 程式設計

5 Games 須用到 Random Number! Why?
NIM Game 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? 蔡文能 C/C++ 程式設計

6 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 Normal distribution? exponential, gamma, Poisson, … 蔡文能 C/C++ 程式設計

7 Turbo C++ 的 rand( )與srand( )
Pseudo random number <stdlib.h> Turbo C++ 的 rand( )與srand( ) #define RAND_MAX 0x7fffu static unsigned long seed=0; int rand( ) { seed = seed * ; return seed % (RAND_MAX+1); } void srand(int newseed) { seed = newseed; static global 變數請參考K&R課本4.6節 就是15個 1的binary 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! 蔡文能 C/C++ 程式設計

8 Unix 上 gcc 的 rand( )與srand( )
Pseudo random number <stdlib.h> Unix 上 gcc 的 rand( )與srand( ) #define RAND_MAX 0x7fffffffu static unsigned long seed=0; int rand( ) { seed = seed * ; return seed % (RAND_MAX+1); } void srand(int newseed) { seed = newseed; static global 變數請參考K&R課本4.6節 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! 蔡文能 C/C++ 程式設計

9 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) 蔡文能 C/C++ 程式設計

10 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 % 蔡文能 C/C++ 程式設計

11 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, . . . 蔡文能 C/C++ 程式設計

12 Uniform Distribution(齊一分配 )
在 發生的機率皆相同 蔡文能 C/C++ 程式設計

13 Normal Distribution (常態分配 )
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14 Standard Normal Distribution (標準常態分配)
平均是 0 標準差 1 蔡文能 C/C++ 程式設計

15 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! 蔡文能 C/C++ 程式設計

16 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。 蔡文能 C/C++ 程式設計

17 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) ? 蔡文能 C/C++ 程式設計

18 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) 蔡文能 C/C++ 程式設計

19 Introduction to simple Statistics
蔡文能 蔡文能 C/C++ 程式設計

20 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" ? 蔡文能 C/C++ 程式設計

21 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 蔡文能 C/C++ 程式設計

22 2005南投縣長選舉大調查 註: 結果是李朝卿當選。
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？ 蔡文能 C/C++ 程式設計

23 2009南投縣長選舉民調 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。
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。 蔡文能 C/C++ 程式設計

24 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%。 蔡文能 C/C++ 程式設計

25 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。 蔡文能 C/C++ 程式設計

26 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。 蔡文能 C/C++ 程式設計

27 1936 Presidential Election and Poll
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28 背景：1936年美國總統選舉 美國經濟正由大蕭條中逐漸恢復 宣稱一：大部分的觀察家認為羅斯福總統將大勝
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。 蔡文能 C/C++ 程式設計

29 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？ 蔡文能 C/C++ 程式設計

30 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 蔡文能 C/C++ 程式設計

31 95%信心水準之下，抽樣誤差在正負3個百分點以內。
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) = 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。 蔡文能 C/C++ 程式設計

32 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:
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! 蔡文能 C/C++ 程式設計

33 Descriptive Statistics
Dispersion Range Standard deviation Variance N Not P (inferential stats) Distribution frequency distribution Histogram (長條圖) Central tendency Mean Median (中位數) mode (眾數) Univariate Analysis Univariate analysis involves the examination across cases of one variable at a time. There are three major characteristics of a single variable that we tend to look at: the distribution , the central tendency and the dispersion N is the number of items in your survey, subjects or cases. 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 蔡文能 C/C++ 程式設計

34 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)? Central Tendency. The central tendency of a distribution is an estimate of the "center" of a distribution of values. There are three major types of estimates of central tendency: Mean Median Mode The Mean or average is probably the most commonly used method of describing central tendency. To compute the mean all you do is add up all the values and divide by the number of values. For example, the mean or average quiz score is determined by summing all the scores and dividing by the number of students taking the exam. For example, consider the test score values: 15, 20, 21, 20, 36, 15, 25, 15 The sum of these 8 values is 167, so the mean is 167/8 = The Median is the score found at the exact middle of the set of values. One way to compute the median is to list all scores in numerical order, and then locate the score in the center of the sample. For example, if there are 500 scores in the list, score #250 would be the median. If we order the 8 scores shown above, we would get: 15,15,15,20,20,21,25,36 There are 8 scores and score #4 and #5 represent the halfway point. Since both of these scores are 20, the median is 20. If the two middle scores had different values, you would have to interpolate to determine the median. The mode is the most frequently occurring value in the set of scores. To determine the mode, you might again order the scores as shown above, and then count each one. The most frequently occurring value is the mode. In our example, the value 15 occurs three times and is the model. In some distributions there is more than one modal value. For instance, in a bimodal distribution there are two values that occur most frequently. Notice that for the same set of 8 scores we got three different values , 20, and for the mean, median and mode respectively. If the distribution is truly normal (i.e., bell-shaped), the mean, median and mode are all equal to each other. 蔡文能 C/C++ 程式設計

35 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 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 Sample Variance 蔡文能 C/C++ 程式設計

36 標準差就是 sqrt (變異數); 阿就是變異數的平方根
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 標準差就是 sqrt (變異數); 阿就是變異數的平方根 蔡文能 C/C++ 程式設計

37 Compute Variance without mean
Variance = (平方和 – 和的平方/n) / n From Wikipedia.org 蔡文能 C/C++ 程式設計

38 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 蔡文能 C/C++ 程式設計

39 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 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. 蔡文能 C/C++ 程式設計

40 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 蔡文能 C/C++ 程式設計

41 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…) 蔡文能 C/C++ 程式設計

42 Normal Distributions and the Standard Deviation
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. 蔡文能 C/C++ 程式設計

43 68-95-99.7 Rule for Normal Curves
68.26% of the observations fall within  of the mean  68.26% + - +3 -3 99.74% +2 -2 95.44% 95.44% of the observations fall within 2 of the mean  99.74% of the observations fall within 3 of the mean  蔡文能 C/C++ 程式設計

44 Notations It is important to distinguish between empirical and theoretical distributions Different notation for each distribution 蔡文能 C/C++ 程式設計

45 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) = 蔡文能 C/C++ 程式設計

46 Confidence Intervals (CI) for µ, from a single sample mean
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47 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)。 蔡文能 C/C++ 程式設計

48 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)。 蔡文能 C/C++ 程式設計

49 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。 蔡文能 C/C++ 程式設計

50 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) 蔡文能 C/C++ 程式設計

51 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 無線網路加解密) WEP : RC4 Stream cipher 蔡文能 C/C++ 程式設計

52 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 蔡文能 C/C++ 程式設計

53 Middle square Jon von Neumann 1946 suggested the production of random number using arithmetic operations of a computer, "middle square", square a previous random number and extract the middle digits, Example generate 10-digit numbers, was , square the next number is "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. 蔡文能 C/C++ 程式設計

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 ? 蔡文能 C/C++ 程式設計

55 RC4 PRNG (1/2) Seeding RC4 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; This is how RC4 is seeded. Every element is touched once as a result of I being incremented across the whole array, however J mightn’t touch every element. 蔡文能 C/C++ 程式設計

56 RC4 PRNG (2/2) rc4byte() { I++; J += S[I]; SWAP(S[I], S[J]);
return (S[ S[I] + S[J] ]); } Byte version First time through this is S[ S[1] + S[S[1]] ]. 蔡文能 C/C++ 程式設計

57 WEP: RC4 加解密 (http://rsa.com)
Pseudo-random number generator Encryption Key K 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 蔡文能 C/C++ 程式設計

58 Games, Random Numbers and Introduction to simple statistics
謝謝捧場 蔡文能 C/C++ 程式設計