題目：
檢索有關下列數學家對統計學發展的貢獻：
- Leonhard Euler (歐拉)
- Friedrich Gauss (高斯)
- Karl Pearson (皮爾森)
- Ronald Aylmer Fisher (費歇)
- Egon Sharpe Pearson (皮爾森)
資料
Euler
積分與概率
歐拉(Euler)積分是其重要貢獻之一,它廣義積 分定義的特殊函數,在概率論與數理統計及數理方程等學科中經常用到
Euler's paper on the game of Rencontre
published in 1753, is E201, "Calcul de la probabilité dans le jeu de rencontre," Mémoires de l'académie de Berlin 7 (1751), 1753, p. 255-270.
Related to above, "Solution quaestionis curiosae ex doctrina combinationum" published in the Mémoires de l'académie des sciences de St. Pétersbourg 3 (1809/10), 1811, p. 57-64.
On lotteries
Euler in correspondence with Frederick the Great and in the papers E338, E412, E600, E812, and E813.
Euler's interest in lotteries
began at the latest in 1749 when he was commissioned by Frederick the Great to render an opinion on a proposed lottery. The first of two letters began 15 September 1749. A second series began on 17 August 1763.
Euler himself wrote several papers prompted by investigations of lotteries.
E812. Read before the Academy of Berlin 10 March 1763 but only published posthumously in 1862. "Reflexions sur une espese singulier de loterie nommée loterie genoise." Opera postuma I, 1862, p. 319-335. The paper determined the probability that a particular number be drawn.
E338. "Sur la probabilité des sequences dans la loterie genoise." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [21] (1765), 1767, p. 191-230. As the name implies, Euler asks for the probability that various sequences of numbers be drawn.
The volume of the journal which contains E338, contains as well a paper by Jean III Bernoulli, "Sur les suites ou séquences dans la loterie de Genes," pp. 234-253 and two papers by Beguelin, "Sur les suites ou séquences dans la lotterie de Gene: First memoir and second memoir, " pp. 231-280.
E412. Read 29 November 1770. "Solution d'une questione tres difficile dans le calcul des probabilités." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [25] (1769), 1771, p. 255-302. This is an analysis of a lottery for which there are several classes and a guaranteed payment.
E600. "Solutio quarundam quaestionum difficiliorum in calculo probabilis." Opuscula Analytica Vol. II, 1785, p. 331-346. Here Euler investigated the probability that all numbers or some fewer numbers be drawn in a sequence of lotteries.
Regarding this latter paper, see De Moivre, 1711, De Mensura Sortis Problem 18 or its nearly identical counterpart in the Doctrine of Chances Problem 39. In these places, de Moivre determined the expectation of one who would cast a die some number of times so as to produce all faces. P.S. Laplace asked for the probability that all tickets will have been withdrawn after a prescribed number of drawings. This problem was solved in "Mémoire sur les suites récurro-récurrentes et sur leurs usages dans la théorie des hasards," Mém. Acad. R. Sci. Paris (Savants étrangers) 6, 1774, pages 353-371. Here Laplace refers to the Genoise Lottery as the Lottery of the Military School. Years later, in the Théorie analytique des Probabilités he asked for the number of drawings for which the probability that all tickets will have come forth is one-half. This is found in Book II, Chapter II, No. 4. The Genoise Lottery is now called the Lottery of France. Jean Trembley, citing the papers of both Euler and Laplace, generalized the solution to the problem in "Recherches sur une question relative au calcul des probabilités," Mémoires de l'Académie royale des sciences et belles-lettres, Berlin 1794/5, pp. 69-108.
E813 "Analyse d'un probleme du calcul des probabilites," Opera Postuma I, 1862, p. 336-341. In this paper, Euler determined the probability that a ticket will be drawn 0, 1, 2, ... times in n successive drawings of r tickets from an urn.
Euler concerned himself with mortality and life expectancy
in E334. "Recherches générales sur la mortalité et la multiplication du genre humain." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [16] (1760), 1797, p.144-164. But see also the section on Life Assurance for the companion piece E335, "Sur les rentes viageres." and the mortality table of Kersseboom.
Life Assurance: E335
E335. "Sur les rentes viageres," Mémoires de l'Académie royale des sciences et belles-lettres de Berlin[16] (1760), 1797, p.165-175. Euler viewed this paper as a continuation of E334. Here he derived a formula to facilitate the computation of a life annuity. Euler observed that there is no advantage to the state to sell annuities where the return is greater than the rate of interest earned by the state. Therefore he proposed the creation of foreborne annuities, purchased, for example, on a child at birth, but due at age 20. This would then permit the accumulation of funds as well as allow the annuity to be offered at a much lower rate.
卡爾·弗里德里希·高斯 Carl Friedrich Gauss
18歲的高斯發現了最小二乘法，并猜測了質數定理。通過對足夠多的測量數據的處理后，可以得到一個新的、概率性質的測量結果。在這些基礎之上，高斯隨后專注于曲面與曲線的計算，并成功得到高斯鐘形曲線(正態分布曲線)。其函數被命名為標準正態分布（或高斯分布），并在概率計算中大量使用。
Gauss's writings on least squares and applications
The collected works of Gauss (Werke) have been published in 12 volumes. These are available through the GDZ: G?ttinger Digitalisierungszentrum. Those relevant to this discussion are the following:
Volume 4 Wahrscheinlichkeitsrechnung und Geometrie
1 ABHANDLUNGEN: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior. Commentationes societatis regiae scientarium Gottingensis recentiores, 5. pp. 33- 62. (1821 Feb. 15) Werke 4, 1-26.
2 Theoria combinationis observationum erroribus minimis obnoxiae: Pars posterior. Commentationes societatis regiae scientarium Gottingensis recentiores, 5. pp. 63-90. (1823 Feb. 2) Werke 4, 27-53.
3 Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. Commentationes societatis regiae scientarium Gottingensis recentiores, 6. pp. 57-98. (1826 Sept. 16) Werke 4 55-94.
4 ANZEIGEN EIGNER ABHANDLUNGEN: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior. G?ttingische gelehrte Anzeigen, 33: 321-327. (1821 Feb. 26) Werke 4, pp. 95-100.
5 Theoria combinationis observationum erroribus minimis obnoxiae: Pars posterior. G?ttingische gelehrte Anzeigen, 32: 313-318. (1823 Feb. 24) Werke 4, pp. 100-104.
6 Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. G?ttingische gelehrte Anzeigen, 153:1521-1527. (1826 Sept. 25) Werke 4, 104-108.
7 AUFSATZ: Bestimmung der Genauigkeit der Beobachtungen Zeitschrift für Astronomie, 1. (1816 March, pp. 185-197) Werke 4, 109-117. (On the Determination of the Precision of Observations)
Volume 6 Astronomische Abhandlungen
[8] Disquisitio de elementis ellipticis Pallidis Commentationes societatis regiae scientarium Gottingensis recentiores, 1. pp. 1-26. (1810) Werke 6, 1-50. (Application of the Method of Least Squares to the Elements of the Planet Pallas) A partial translation into German with comments is given as "über die elliptischen Elemente der Pallas," Monatliche Correspondenz, 1811, Vol. XXIV, pp. 449-465.
[9] Chronometrische L?ngenbestimmungen Astronomische Nachrichten Band 5, S. 227-240 (1826 Nov.). Werke 6, 455-458. (On the Chronometric Determination of Longitude)
Volume 7 Theoretische Astronomie
[10] Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. Perthes and Besser, Hamburg. (1809) Reprinted in Werke 7, pp 1-261.
Volume 9 Geod?sie. Fortsetzung von Band 4.
[11] Bestimmung des Breitenunterschiedes zwischen den Sternwarten von G?ttingen und Altona durch Beobachtungen am Ramsden'schen Zenithsector (1828) Werke 9, 1-64.
[12] Anwendungen der Wahrscheinlichkeitsrechnung auf eine Aufgabe der praktischen Geometrie Astronomische Nachrichten 1, S. 81-86. (1823) Werke 9, 231-237 (Application of Calculus of Probabilities to Practical Geometry)
In addition to this, Gauss's work on "recursive least squares" estimation, or "recursive updating" (Section 13) has only recently been noticed (or sometimes rediscovered) and applied; ［Sprott, David A. (1978) Gauss's contributions to statistics. Historia Mathematica 5: 183-203.］
卡爾·皮爾遜 Karl Pearson
an influential English mathematician who has been credited with establishing the discipline of mathematical statistics.[3
Pearson's work was all-embracing in the wide application and development of mathematical statistics, and encompassed the fields of biology, epidemiology, anthropometry, medicine and social history. In 1901, with Weldon and Galton, he founded the journal Biometrika whose object was the development of statistical theory. He edited this journal until his death. Among those who assisted Pearson in his research were a number of female mathematicians who included Beatrice Mabel Cave-Browne-Cave and Frances Cave-Browne-Cave. He also founded the journal Annals of Eugenics (now Annals of Human Genetics) in 1925. He published the Drapers' Company Research Memoirs largely to provide a record of the output of the Department of Applied Statistics not published elsewhere.
Pearson's thinking underpins many of the 'classical' statistical methods which are in common use today. Examples of his contributions are:
Correlation coefficient. The correlation coefficient (first conceived by Francis Galton) was defined as a product-moment, and its relationship with linear regression was studied.[11]
Method of moments. Pearson introduced moments, a concept borrowed from physics, as descriptive statistics and for the fitting of distributions to samples.
Pearson's system of continuous curves. A system of continuous univariate probability distributions that came to form the basis of the now conventional continuous probability distributions. Since the system is complete up to the fourth moment, it is a powerful complement to the Pearsonian method of moments.
Chi distance. A precursor and special case of the Mahalanobis distance.[12]
P-value. Defined as the probability measure of the complement of the ball with the hypothesized value as center point and chi distance as radius.[12]
Foundations of the statistical hypothesis testing theory and the statistical decision theory.[12] In the seminal "On the criterion..." paper,[12] Pearson proposed testing the validity of hypothesized values by evaluating the chi distance between the hypothesized and the empirically observed values via the p-value, which was proposed in the same paper. The use of preset evidence criteria, so called alpha type-I error probabilities, was later proposed by Jerzy Neyman and Egon Pearson.[13]
Pearson's chi-squared test. A hypothesis test using normal approximation for discrete data.
Principal component analysis. The method of fitting a linear subspace to multivariate data by minimizing the chi distances.[14][15]
In the course of his studies of race, Pearson devised a Coefficient of Racial Likeness, calculated from several measurements of the human skull.
[http://en.wikipedia.org/wiki/Karl_Pearson#Contributions_to_statistics]
羅納德·愛爾默·費雪 Ronald Aylmer Fisher
一名英國統計學家、演化生物學家與遺傳學家。他是現代統計學與現代演化論的奠基者之一
同時在這段期間，他也發表了許多與生物統計相關的論文，包括《孟德爾遺傳假定下的親戚之間的相關性》（The Correlation Between Relatives on the Supposition of Mendelian Inheritance）。這篇論文在1916年完成，並在1918年發表，它同時建立了以生物統計為基礎的遺傳學，以及著名的統計學分法變異數分析（analysis of variance，簡寫為ANOVA，也稱方差分析）。(Fisher is known as one of the chief architects of the neo-Darwinian synthesis, for his important contributions to statistics, including the analysis of variance (ANOVA), method of maximum likelihood, fiducial inference, and the derivation of various sampling distributions, and for being one of the three principal founders of population genetics.)
接下來的幾年中，費雪開始構想新的統計方法，如實驗設計法（design of experiments）。1925年，他的第一本書出版，書名為《研究者的統計方法》（Statistical Methods for Research Workers）[8]。到了1935年，延續本書的《實驗設計》（The Design of Experiments）出版。兩本書建立了實驗設計法的基礎，並受到多次翻譯與再版。
除了新的統計方法，費雪也將先前的變異數分析研究進行補強與修飾，因而發明出最大似然估計，並發展出充分性（sufficiency）、輔助統計、費雪線性判別（Fisher's linear discriminator）與費雪資訊(也譯為費希爾信息)（Fisher information）等統計概念。
(卡爾·皮爾遜繼法蘭西斯·高爾頓之后，發展了回歸與相關的理論，得到母體的概念，并認為統計研究不是樣本本身，而是根據樣本對母體的推斷。由此導出了擬合優度檢驗：即作為樣本取出的若干個體是否擬合從理論上所確定的母體分布問題。
1894年，他提出了矩估計法，并在此后發展了這一方法。
1900年，他創立和發展了卡方檢定的理論，在理論分布完全給定的情況下，給出了擬合優度檢驗的卡方統計量的極限定理。
他考察一些生物學方面數據后，發現不少分布與正態分布呈明顯偏倚，創立了概率密度函數族，……他是從數學上對生物統計研究的第一人，1901年他與高爾頓、韋爾登一起，創辦了“生物統計學”雜志，使生物統計學有了自己的一席之地）
埃根 皮爾遜 Egon Sharpe Pearson
Pearson is best known for development of the Neyman-Pearson lemma of statistical hypothesis testing.
and responsible for many important contributions to problems of statistical inference and methodology, especially in the development and use of the likelihood ratio criterion. Has played a leading role in furthering the applications of statistical methods — for example, in industry, and also during and since the war, in the assessment and testing of weapons
上交內容
一些數學家對統計學發展的貢獻——高斯、歐拉、費歇以及皮爾遜父子
發展自 17 世紀中葉，統計學作為概率論與數理統計學的應用，其方法的可靠性與準確性均為背后龐大的數學基礎支持著。下文便介紹五位在統計學發展歷史上有一定貢獻的數學家。
萊昂哈德·歐拉
歐拉（Leonhard Euler）是有史以來最重要的數學家之一，他在數論、拓撲學、圖論等多個數學領域——特別是微積分——上所作的貢獻必然直接或間接地對現代統計學的形成有不可忽視的影響。其中包括了歐拉積分，它廣義積分定義的特殊函數常被應用在概率論與數理統計及數理方程等學科中。1
此外，歐拉還曾發表過多個直接涉及概率與統計方面問題的研究，如《重遇游戲中概率的計算》（"Calcul de la probabilité dans le jeu de rencontre"）、《關于一種名為 Genoise 的彩票的思考》（"Reflexions sur une espese singulier de loterie nommée loterie genoise."）和《論人壽保險》（"Sur les rentes viageres"）等，均與概率的計算以及應用有關。2
卡爾·弗里德里希·高斯
高斯（Carl Friedrich Gauss），如同歐拉，是歷史上最重要的數學家之一，他在幾何、數學分析等方面有極大貢獻；但相對歐拉，高斯對概率與數理統計的影響更直接顯著。
高斯發現了統計學中極為基本的最小二乘法，并對其應用有大量的研究。從此人們有了通過對足夠多的測量數據的處理來得到一個新的、概率性質的測量結果的方法。2此外，他還發展出遞推最小二乘法估計（"recursive least squares" estimation）。3
其后，高斯在曲面與曲線的計算中得到了正態分布曲線以及其函數（高斯分布），一個在統計學的許多方面有重大影響力的概率分布。4
卡爾·皮爾遜
卡爾·皮爾遜（Karl Pearson）是被譽為數理統計學創始人的極有影響力的數學家。他的研究涵蓋了廣泛的數理統計學發展與應用領域，包括生物學，人類學流行病學和社會科學等。5
他參與多種統計學期刊的創始和編輯，包括《Biometrika》、《Annals of Human Genetics》。他的研究為后世留下了大量經典的統計學方法，如皮爾森相關系數（Pearson product-moment correlation coefficient）、p值（p-value）、皮爾森分布（Pearson distribution）、皮爾森卡方檢定（Pearson's chi-squared test）、主成分分析，以及假設檢驗的理論基礎。5
羅納德·費歇
英國統計學家費歇（Ronald Aylmer Fisher）對統計學的影響如此大，以至于被稱為“幾乎一個人創立了現代統計學的天才”。6
他是“從數學上對生物統計研究的第一人”，發表過許多與生物統計學有關的研究，如《孟德爾遺傳假設下的親緣相關性》（The Correlation Between Relatives on the Supposition of Mendelian Inheritance）、《自然選擇中的基因理論》（The Genetical Theory of Natural Selection）。而他建立的普遍的統計學方法則有方差分析（analysis of variance）、實驗設計（design of experiments）、線性判別分析（Linear Discriminant Analysis）、費歇精確檢驗（Fisher exact test）等等，并發展出充分性、輔助統計、費歇信息等統計概念。6
埃根·皮爾遜
卡爾·皮爾遜之子埃根·皮爾遜（Egon Sharpe Pearson）在統計學上最著名的貢獻便是發展出奈曼–皮爾森引理。此外，他的重要工作也涉及統計推斷和統計方法等方面的問題，他在推廣統計方法實踐上也走在前沿。7
Reference
- "Leonhard Euler" on Wikipedia
- Sources in the History of Probability and Statistics
- Sprott, David A. (1978) Gauss's contributions to statistics. Historia Mathematica 5: 183-203.
- "Carl Friedrich Gauss" on Wikipedia
- "Karl Pearson" on Wikipedia
- "Ronald Fisher" on Wikipedia
- "Egon Pearson" on Wikipedia
