题目：
检索有关下列数学家对统计学发展的贡献：
- Leonhard Euler (欧拉)
- Friedrich Gauss (高斯)
- Karl Pearson (皮尔森)
- Ronald Aylmer Fisher (费歇)
- Egon Sharpe Pearson (皮尔森)
资料
Euler
积分与概率
欧拉(Euler)积分是其重要贡献之一,它广义积 分定义的特殊函数,在概率论与数理统计及数理方程等学科中经常用到
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岁的高斯发现了最小二乘法，并猜测了質数定理。通过对足够多的测量数据的处理后，可以得到一个新的、概率性质的测量结果。在这些基础之上，高斯随后專注于曲面与曲线的计算，并成功得到高斯钟形曲线(正态分布曲线)。其函数被命名为标准正态分布（或高斯分布），并在概率计算中大量使用。
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: Göttinger 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. Göttingische gelehrte Anzeigen, 33: 321-327. (1821 Feb. 26) Werke 4, pp. 95-100.
5 Theoria combinationis observationum erroribus minimis obnoxiae: Pars posterior. Göttingische gelehrte Anzeigen, 32: 313-318. (1823 Feb. 24) Werke 4, pp. 100-104.
6 Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. Göttingische 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 Längenbestimmungen 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 Geodäsie. Fortsetzung von Band 4.
[11] Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen 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.］
卡尔·皮尔逊 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(也译为费希尔信息)（Fisher information）等統計概念。
(卡尔·皮尔逊继法兰西斯·高尔顿之后，发展了回归与相关的理论，得到母體的概念，并认为统计研究不是样本本身，而是根据样本对母體的推断。由此導出了拟合优度检验：即作为样本取出的若干个体是否拟合从理论上所确定的母體分布问题。
1894年，他提出了矩估计法，并在此后发展了这一方法。
1900年，他创立和发展了卡方檢定的理论，在理论分布完全给定的情况下，给出了拟合优度检验的卡方統計量的极限定理。
他考察一些生物学方面数据后，发现不少分布与正态分布呈明显偏倚，创立了概率密度函数族，……他是从数学上对生物统计研究的第一人，1901年他与高尔顿、韦尔登一起，创办了“生物统计学”杂志，使生物统计学有了自己的一席之地）
埃根 皮尔逊 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
上交内容
一些数学家对统计学发展的贡献——高斯、欧拉、费歇以及皮尔逊父子
发展自 17 世纪中叶，统计学作为概率论与数理统计学的应用，其方法的可靠性与准确性均为背后庞大的数学基础支持着。下文便介绍五位在统计学发展历史上有一定贡献的数学家。
莱昂哈德·欧拉
欧拉（Leonhard Euler）是有史以来最重要的数学家之一，他在数论、拓扑学、图论等多个数学领域——特别是微积分——上所作的贡献必然直接或间接地对现代统计学的形成有不可忽视的影响。其中包括了欧拉积分，它广义积分定义的特殊函数常被应用在概率论与数理统计及数理方程等学科中。1
此外，欧拉还曾发表过多个直接涉及概率与统计方面问题的研究，如《重遇游戏中概率的计算》（"Calcul de la probabilité dans le jeu de rencontre"）、《关于一种名为 Genoise 的彩票的思考》（"Reflexions sur une espese singulier de loterie nommée loterie genoise."）和《论人寿保险》（"Sur les rentes viageres"）等，均与概率的计算以及应用有关。2
卡尔·弗里德里希·高斯
高斯（Carl Friedrich Gauss），如同欧拉，是历史上最重要的数学家之一，他在几何、数学分析等方面有极大贡献；但相对欧拉，高斯对概率与数理统计的影响更直接显著。
高斯发现了统计学中极为基本的最小二乘法，并对其应用有大量的研究。从此人们有了通过对足够多的测量数据的处理来得到一个新的、概率性质的测量结果的方法。2此外，他还发展出递推最小二乘法估计（"recursive least squares" estimation）。3
其后，高斯在曲面与曲线的计算中得到了正态分布曲线以及其函数（高斯分布），一个在统计学的许多方面有重大影响力的概率分布。4
卡尔·皮尔逊
卡尔·皮尔逊（Karl Pearson）是被誉为数理统计学创始人的极有影响力的数学家。他的研究涵盖了广泛的数理统计学发展与应用领域，包括生物学，人类学流行病学和社会科学等。5
他参与多种统计学期刊的创始和编辑，包括《Biometrika》、《Annals of Human Genetics》。他的研究为后世留下了大量经典的统计学方法，如皮尔森相关系数（Pearson product-moment correlation coefficient）、p值（p-value）、皮尔森分布（Pearson distribution）、皮尔森卡方检定（Pearson's chi-squared test）、主成分分析，以及假设检验的理论基础。5
罗纳德·费歇
英国统计学家费歇（Ronald Aylmer Fisher）对统计学的影响如此大，以至于被称为“几乎一个人创立了现代统计学的天才”。6
他是“从数学上对生物统计研究的第一人”，发表过许多与生物统计学有关的研究，如《孟德尔遗传假设下的亲缘相关性》（The Correlation Between Relatives on the Supposition of Mendelian Inheritance）、《自然选择中的基因理论》（The Genetical Theory of Natural Selection）。而他建立的普遍的统计学方法则有方差分析（analysis of variance）、实验设计（design of experiments）、线性判别分析（Linear Discriminant Analysis）、费歇精确检验（Fisher exact test）等等，并发展出充分性、辅助统计、费歇信息等统计概念。6
埃根·皮尔逊
卡尔·皮尔逊之子埃根·皮尔逊（Egon Sharpe Pearson）在统计学上最著名的贡献便是发展出奈曼–皮尔森引理。此外，他的重要工作也涉及统计推断和统计方法等方面的问题，他在推广统计方法实践上也走在前沿。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
