# 機率思想的發展 機率思想的發展

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### 機率思想的發展 機率思想的發展

http://

http://www.math.ntu.edu.tw/~shiehnrwww.math.ntu.edu.tw/~shiehnr

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### Part I Timeline 時間線 時間線

 From Alpha to Omega 隨機性在早期西方文明From Alpha to Omega 隨機性在早期西方文明

 Light in the Dark Age 黑暗時代中的修道院與大學Light in the Dark Age 黑暗時代中的修道院與大學

 From Omega to Alpha From Omega to Alpha 文藝復興文藝復興: : 屬地智慧的發端屬地智慧的發端

 Fermat and Pascal Fermat and Pascal 費馬與巴斯卡的通信費馬與巴斯卡的通信

 Gauss and Laplace Gauss and Laplace 高斯與拉普拉斯的榮耀與高斯與拉普拉斯的榮耀與 KusoKuso

 Gauss vsGauss vs Cauchy distribution “Cauchy distribution 正規正規 與與 “非正規非正規

 Poisson and Poisson process 泊松過程Poisson and Poisson process 泊松過程: : 等待等待

 Brownian motion and Wiener process 布朗運動與維納過Brownian motion and Wiener process 布朗運動與維納過 程程: : 生命力生命力與與 分子互動分子互動

 Mathematical Probability Theory Mathematical Probability Theory 公理化機率論公理化機率論

 Probability in the 21Probability in the 21stst CenturyCentury

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###   .. .. 你們要去使萬民作我的門徒 你們要去使萬民作我的門徒 … … ( ( 太 太 28:19) 28:19)

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left: Fortuna( a goddess), right: an

left: Fortuna( a goddess), right: an astragalusastragalus dicedice

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## Light in the Dark Age Light in the Dark Age

### 黑暗時代中的修道院與大學 黑暗時代中的修道院與大學

 The collapse of Rome (455 A.D.)The collapse of Rome (455 A.D.)

 教會, 教會, 修道院修道院, , 與與 大學大學

 The light shines in the darkness, but the darkness has not The light shines in the darkness, but the darkness has not understood it (John 1:5)

understood it (John 1:5)

 Alcuin of York ( 735Alcuin of York ( 735--804) and Charlemagne (742804) and Charlemagne (742--814 ) 814 )

 Leonardo the Leonardo the PisanPisan (Fibonacci, 1170(Fibonacci, 1170--1250)1250)

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### 文藝復興 文藝復興 : : 屬地智慧的發端 屬地智慧的發端

 The Renaissance The Renaissance 文藝復興文藝復興

 With the Greeks there is on the whole little speculation With the Greeks there is on the whole little speculation about the natural phenomena of the earth (as opposed to about the natural phenomena of the earth (as opposed to the heavens) which is based on empirical hypotheses. With the heavens) which is based on empirical hypotheses. With the renaissance of human intellect there is the beginning of the renaissance of human intellect there is the beginning of the empirical tradition, from observation to hypothesis and the empirical tradition, from observation to hypothesis and back to observation again, which has led to the great

back to observation again, which has led to the great strides of modern science (F.N. David 1962).

strides of modern science (F.N. David 1962).

 敬畏耶和華是知識的開端敬畏耶和華是知識的開端 ( ( 箴言箴言 1:7 )1:7 )

 敬畏耶和華是智慧的開端敬畏耶和華是智慧的開端 ( ( 箴言箴言 9:10)9:10)

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### 費馬與巴斯卡的通信 費馬與巴斯卡的通信

 Pierre de Fermat (1601Pierre de Fermat (1601--1665)1665)

 BlaiseBlaise Pascal (1623Pascal (1623--1662)1662)

 Rev. Father M. Rev. Father M. MersenneMersenne’’ss ““AcademyAcademy””, forerunner , forerunner of Royal Society of London (1660) and

of Royal Society of London (1660) and AcademieAcademie des Sciences de Paris (1665)

des Sciences de Paris (1665)

 In 1654, they have had a series of letters on In 1654, they have had a series of letters on

““problem of pointsproblem of points””

 一般認為這是一般認為這是 機率論機率論 的的 緣起緣起 ,但,但

 賭注分配問題先前已出現於賭注分配問題先前已出現於 FraFra Luca Luca PacioloPaciolo (1445(1445--1509)1509)的的 1494

1494 書中書中

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thth

###   甲贏 甲贏 1 1 局乙贏 局乙贏 0 0 局 局 : 44 : 20 : 44 : 20

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Fermat, Pascal, PacioloFermat, Pascal, Paciolo (from left to right)

(from left to right)

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### 高斯與拉普拉斯的榮耀與 Kuso Kuso

 Carl Friedrich Gauss (1777Carl Friedrich Gauss (1777--1855) 1855) 數學王子數學王子

 Pierre-Pierre-Simon Laplace (1749Simon Laplace (1749--1827) 法國牛頓1827) 法國牛頓



 Gauss: theory of errors and method of least squares Gauss: theory of errors and method of least squares (1795,1809)

(1795,1809)

 Laplace: Analytic Theory of ProbabilitiesLaplace: Analytic Theory of Probabilities (1812)(1812)

 前者一般認為是統計學的發源前者一般認為是統計學的發源

 後者一般認為是機率論的大成(up to then)後者一般認為是機率論的大成(up to then)

 但兩位都有但兩位都有 KusoKuso 名言名言 : 參考: 參考 E. T. Bell: Men of E. T. Bell: Men of Mathematics (

Mathematics (中譯中譯: : 大數學家大數學家, , 九章九章) )

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### “ “ Normal Normal ” ” distribution, correct and unbeatable? distribution, correct and unbeatable?

 each of each of ““n particlesn particles”” contributes contributes ““1/n to the whole1/n to the whole””, so , so that

that ““ the middlethe middle”” plays normal and deciding role. Yet, it plays normal and deciding role. Yet, it has been observed that

has been observed that ““power lawpower law”” and and ““heavy tailheavy tail””

phenomena appear both in earthly and heavenly sciences phenomena appear both in earthly and heavenly sciences

and wisdoms. Compare the curves below with those in and wisdoms. Compare the curves below with those in

the next slide the next slide

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Cauchy, Gauss, Laplace, Poisson (from left to right) Cauchy, Gauss, Laplace, Poisson (from left to right)

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### 泊松過程 泊松過程 : : 等待 等待

 Simeon-Simeon-Denis Poisson (1781Denis Poisson (1781--18401840))

 On the probability of the judgments of course of justice (1837)On the probability of the judgments of course of justice (1837)

 Poisson distribution : 以下為機率導論一考題Poisson distribution : 以下為機率導論一考題

 Ms TW 打字錯誤個數視為某離散機率分佈Ms TW 打字錯誤個數視為某離散機率分佈..已知平均而言每已知平均而言每

 20字中有20字中有11個錯字個錯字. . 利用利用 CLT CLT 估計估計 Ms TW 打Ms TW 4500 字文章4500 字文章 錯字錯字 $<200$ $<200$

 Poisson process : waiting for coming, 以下為隨機過程導論一考題Poisson process : waiting for coming, 以下為隨機過程導論一考題

 Let $X$ be the Poisson process with parameter $\Let$X$be the Poisson process with parameter$\lambda$. Let$T_ilambda$. Let$T_i, , i=1,2,

i=1,2,\\cdots$be the sequence of intercdots$ be the sequence of inter——arrival times, prove that it is arrival times, prove that it is iidiid with $Exp( with$Exp(\\lambda)$as common distribution. Derive the distribution of lambda)$ as common distribution. Derive the distribution of the total waiting time until $the total waiting time until$nn--$th$th event, i.e. $S_nevent, i.e.$S_n=T_1+\=T_1+\cdots+T_n$. cdots+T_n$.

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### 布朗運動與維納過程 : : ““ 生命力 生命力 ”” 與 與 ““ 分子互動 分子互動 ””

 Robert Brown (1773Robert Brown (1773--1858) 1858) 觀察觀察

 Norbert Wiener (1894-Norbert Wiener (1894-1964) 1964) 建模建模

 顯微鏡下花粉的運動顯微鏡下花粉的運動

 愛因斯坦的物理學家觀點愛因斯坦的物理學家觀點(1906, Brownian motion (1906, Brownian motion and molecular movement),

and molecular movement),一般將此與光電現象及一般將此與光電現象及 相對論併列為愛因斯坦三大學術成就



 愛因斯坦的愛因斯坦的 KusoKuso 名言名言 ::

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### 公理化機率論 公理化機率論

 A.N. KolmogorovA.N. Kolmogorov ( 1903-( 1903-1987)1987)

 Foundation of Probability Theory (1933)Foundation of Probability Theory (1933)

 The sample space ΩThe sample space Ω, is a nonempty , is a nonempty setset whose elements are known as whose elements are known as outcomes or states of nature and are often given the symbol

outcomes or states of nature and are often given the symbol ωω. The set . The set of all the possible outcomes of an experiment is known as the sa

of all the possible outcomes of an experiment is known as the sample mple space of the experiment.

space of the experiment.

 The probability measure PThe probability measure P is a function from the sigma-is a function from the sigma-algebra of algebra of events to the real numbers that assigns to each event a

events to the real numbers that assigns to each event a probabilityprobability between 0 and 1. It must satisfy the

between 0 and 1. It must satisfy the probability axiomsprobability axioms

 A random variable X is a measurable mapping from ωA random variable X is a measurable mapping from ω in in ΩΩ to X(to X(ωω) in ) in R(real

R(real number system).number system).

 THE WORDS OF OMEGASTHE WORDS OF OMEGAS

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stst

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### 機率論的迷思 I I

 以下引自以下引自 Richard von Richard von MisesMises: Probability, Statistics : Probability, Statistics and Truth, Lecture One ( 1950, p. 9)

and Truth, Lecture One ( 1950, p. 9)

 We do not deal with the problem of the historical We do not deal with the problem of the historical accuracy of Biblical narratives

accuracy of Biblical narratives…… Similarly we do Similarly we do not concern ourselves with any of those problems not concern ourselves with any of those problems of the moral sciences

of the moral sciences ……. The unlimited extension . The unlimited extension of the validity of the exact sciences was a

of the validity of the exact sciences was a characteristic feature of the exaggerated characteristic feature of the exaggerated

rationalism of the 18th century. We do not intend rationalism of the 18th century. We do not intend

to commit the same mistake.

to commit the same mistake.

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### 機率論的迷思 II II

 以下亦引自以下亦引自 Richard von Richard von MisesMises: Probability, : Probability,

Statistics and Truth, Lecture One ( 1950, p 17) Statistics and Truth, Lecture One ( 1950, p 17)

 Similarly, the insurance mathematicians (working Similarly, the insurance mathematicians (working for some insurance company)

for some insurance company)…… use a figure such use a figure such as the above 0.011 (company

as the above 0.011 (company’’s death probability s death probability of a registered person with age 40 dies at 41)

of a registered person with age 40 dies at 41)…It …It is utter nonsense to say, for instance, that Mr. X, is utter nonsense to say, for instance, that Mr. X,

now aged 40, has the probability 0.011 of dying in now aged 40, has the probability 0.011 of dying in

the course of the next year.

the course of the next year.

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### Probability in ComplexityProbability in Complexity

 以下引自以下引自 Thomas Thomas LuxLux: J. Econ. : J. Econ. BehBeh. & Org. vol. . & Org. vol.

33 (1998) The socio

33 (1998) The socio--economic dynamics of economic dynamics of speculative markets, p144.

speculative markets, p144.

 It has been highly debated, whether the apparent It has been highly debated, whether the apparent deviation of the (highly complicated) data from a deviation of the (highly complicated) data from a random walk is due to stochastic or deterministic random walk is due to stochastic or deterministic nonnon--linearitieslinearities……....

 金融市場不是丟銅板金融市場不是丟銅板, , 想當然恶想當然恶,,但數據的高複雜性但數據的高複雜性 由何而由何而 來來 ??

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## Abraham de Moivre Moivre ( ( 隸美弗 隸美弗 ) )

 蒙福高壽蒙福高壽 16671667-1754 ( -1754 ( 牛頓牛頓 1643-1643-1727 )1727 )

 因信仰流亡異國因信仰流亡異國 ( 自( 自1688 1688 就就 客居英國終生客居英國終生 ))

 結識牛頓結識牛頓 受其賞識受其賞識 獲選入英國皇家學會獲選入英國皇家學會

 找不到正式教職找不到正式教職 以家教為生以家教為生

 長期在一間咖啡店為人計算賭注以補貼生活長期在一間咖啡店為人計算賭注以補貼生活

 計算得投銅板的中央極限定理計算得投銅板的中央極限定理

 出版出版 Doctrine of Chances (1st Doctrine of Chances (1st eded 1718, 2nd ed1718, 2nd ed 1738, 3rd 1738, 3rd eded 1756(posthumous) )1756(posthumous) )

 在書的第三版卷末寫下信仰見證在書的第三版卷末寫下信仰見證

 書的內容被人摽竊書的內容被人摽竊 書的學術榮耀被書的學術榮耀被 Laplace Laplace 取代取代

 但尚有但尚有 F. N. (Florence Nightingale) David F. N. (Florence Nightingale) David 為他為他 專專 章寫章寫 下下 以上人生以上人生 供讀者低迴長思供讀者低迴長思 ( F.N. David: Games, gods, and ( F.N. David: Games, gods, and gambling, a history of probability and statistical ideas,

gambling, a history of probability and statistical ideas, 1962, republished by Dover 1998)

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Abraham de

Abraham de MoivreMoivre

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### 觸摸神 觸摸神

 神賜予我們智慧與意志去探索祂神賜予我們智慧與意志去探索祂

 數學數學 ( mathematics)( mathematics)

 科學科學 (sciences)(sciences)

 哲學哲學 (philosophy)(philosophy)

 神學神學 (theology)(theology)

 這麼個這麼個 from Alpha to Omega 的過程我們是否以from Alpha to Omega 的過程我們是否以

 謙卑感恩底謙卑感恩底 Walk along with HimWalk along with Him

 For the Lord gives wisdom; For the Lord gives wisdom;

 from His mouth come knowledge and understandingfrom His mouth come knowledge and understanding

 (Proverbs 2:6)(Proverbs 2:6)

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