Preface xvi Acknowledgments xix Introduction: LANDMARKS IN PRE-LAPLACEAN STATISTICS xx PART ONE: LAPLACE 1 1 The Laplacean Revolution 3 1.1 Pierre?]Simon de Laplace (1749-1827) 3 1.2 Laplace''s Work in Probability and Statistics 7 1.2.1 "Mémoire sur les suites récurro?]récurrentes" (1774): Definition of Probability 7 1.2.2 "Mémoire sur la probabilité des causes par les événements" (1774) 9 1.2.

2.1 Bayes'' Theorem 9 1.2.2.2 Rule of Succession 13 1.2.2.3 Proof of Inverse Bernoulli Law.

Method of Asymptotic Approximation. Central Limit Theorem for Posterior Distribution. Indirect Evaluation of et2 0 dt 14 1.2.2.4 Problem of Points 18 1.2.2.

5 First Law of Error 19 1.2.2.6 Principle of Insufficient Reason (Indifference) 24 1.2.2.7 Conclusion 25 1.2.

3 "Recherches sur l''intégration des équations différentielles aux différences finis" (1776) 25 1.2.3.1 Integration of Difference Equations. Problem of Points 25 1.2.3.2 Moral Expectation.

On d''Alembert 26 1.2.4 "Mémoire sur l''inclinaison moyenne des orbites" (1776): Distribution of Finite Sums, Test of Significance 28 1.2.5 "Recherches sur le milieu qu''il faut choisir entre les resultants de plusieurs observations" (1777): Derivation of Double Logarithmic Law of Error 35 1.2.6 "Mémoire sur les probabilités" (1781) 42 1.2.

6.1 Introduction 42 1.2.6.2 Double Logarithmic Law of Error 44 1.2.6.3 Definition of Conditional Probability.

Proof of Bayes'' Theorem 46 1.2.6.4 Proof of Inverse Bernoulli Law Refined 50 1.2.6.5 Method of Asymptotic Approximation Refined 53 1.2.

6.6 Stirling''s Formula 58 1.2.6.7 Direct Evaluation of e t2 0 dt 59 1.2.6.8 Theory of Errors 60 1.

2.7 "Mémoire sur les suites" (1782) 62 1.2.7.1 De Moivre and Generating Functions 62 1.2.7.2 Lagrange''s Calculus of Operations as an Impetus for Laplace''s Generating Functions 65 1.

2.8 "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres" (1785) 70 1.2.8.1 Method of Asymptotic Approximation Revisited 70 1.2.8.2 Stirling''s Formula Revisited 73 1.

2.8.3 Genesis of Characteristic Functions 74 1.2.9 "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres (suite)" (1786): Philosophy of Probability and Universal Determinism, Recognition of Need for Normal Probability Tables 78 1.2.10 "Sur les naissances" (1786): Solution of the Problem of Births by Using Inverse Probability 79 1.2.

11 "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres et sur leur application aux probabilités" (1810): Second Phase of Laplace''s Statistical Career, Laplace''s First Proof of the Central Limit Theorem 83 1.2.12 "Supplément au Mémoire sur les approximations des formules qui sont fonctions de très grands nombres et sur leur application aux probabilités" (1810): Justification of Least Squares Based on Inverse Probability, The Gauss-Laplace Synthesis 90 1.2.13 "Mémoire sur les intégrales définies et leur applications aux probabilités, et spécialement à la recherche du milieu qu''il faut choisir entre les résultats des observations" (1811): Laplace''s Justification of Least Squares Based on Direct Probability 90 1.2.14 Théorie Analytique des Probabilités (1812): The de Moivre-Laplace Theorem 90 1.2.

15 Laplace''s Probability Books 92 1.2.15.1 Théorie Analytique des Probabilités (1812) 92 1.2.15.2 Essai Philosophique sur les Probabilités (1814) 95 1.3 The Principle of Indifference 98 1.

3.1 Introduction 98 1.3.2 Bayes'' Postulate 99 1.3.3 Laplace''s Rule of Succession. Hume''s Problem of Induction 102 1.3.

4 Bertrand''s and Other Paradoxes 106 1.3.5 Invariance 108 1.4 Fourier Transforms, Characteristic Functions, and Central Limit Theorems 113 1.4.1 The Fourier Transform: From Taylor to Fourier 114 1.4.2 Laplace''s Fourier Transforms of 1809 120 1.

4.3 Laplace''s Use of the Fourier Transform to Solve a Differential Equation (1810) 122 1.4.4 Lagrange''s 1776 Paper: A Precursor to the Characteristic Function 123 1.4.5 The Concept of Characteristic Function Introduced: Laplace in 1785 127 1.4.6 Laplace''s Use of the Characteristic Function in his First Proof of the Central Limit Theorem (1810) 128 1.

4.7 Characteristic Function of the Cauchy Distribution: Laplace in 1811 128 1.4.8 Characteristic Function of the Cauchy Distribution: Poisson in 1811 131 1.4.9 Poisson''s Use of the Characteristic Function in his First Proof of the Central Limit Theorem (1824) 134 1.4.10 Poisson''s Identification of the Cauchy Distribution (1824) 138 1.

4.11 First Modern Rigorous Proof of the Central Limit Theorem: Lyapunov in 1901 139 1.4.12 Further Extensions: Lindeberg (1922), Lévy (1925), and Feller (1935) 148 1.5 Least Squares and the Normal Distribution 149 1.5.1 First Publication of the Method of Least Squares: Legendre in 1805 149 1.5.

2 Adrain''s Research Concerning the Probabilities of Errors (1808): Two Proofs of the Normal Law 152 1.5.3 Gauss'' First Justification of the Principle of Least Squares (1809) 159 1.5.3.1 Gauss'' Life 159 1.5.3.

2 Derivation of the Normal Law. Postulate of the Arithmetic Mean 159 1.5.3.3 Priority Dispute with Legendre 163 1.5.4 Laplace in 1810: Justification of Least Squares Based on Inverse Probability, the Gauss-Laplace Synthesis 166 1.5.

5 Laplace''s Justification of Least Squares Based on Direct Probability (1811) 169 1.5.6 Gauss'' Second Justification of the Principle of Least Squares in 1823: The Gauss-Markov Theorem 177 1.5.7 Hagen''s Hypothesis of Elementary Errors (1837) 182 PART TWO : FROM GALTON TO FISHER 185 2 Galton, Regression, and Correlation 187 2.1 Francis Galton (1822-1911) 187 2.2 Genesis of Regression and Correlation 190 2.2.

1 Galton''s 1877 Paper, "Typical Laws of Heredity": Reversion 190 2.2.2 Galton''s Quincunx (1873) 195 2.2.3 Galton''s 1885 Presidential Lecture and Subsequent Related Papers: Regression, Discovery of the Bivariate Normal Surface 197 2.2.4 First Appearance of Correlation (1888) 206 *2.2.

5 Some Results on Regression Based on the Bivariate Normal Distribution: Regression to the Mean Mathematically Explained 209 2.2.5.1 Basic Results Based on the Bivariate Normal Distribution 209 2.2.5.2 Regression to the Mean Mathematically Explained 211 2.3 Further Developments after Galton 211 2.

3.1 Weldon (1890; 1892; 1893) 211 2.3.2 Edgeworth in 1892: First Systematic Study of the Multivariate Normal Distribution 213 2.3.3 O rigin of Pearson''s r (Pearson et al. 1896) 220 2.3.

4 Standard Error of r (Pearson et al. 1896; Pearson and Filon 1898; Student 1908; Soper 1913) 224 2.3.5 Development of Multiple Regression, Galton''s Law of Ancestral Heredity, First Explicit Derivation of the Multivariate Normal Distribution (Pearson et al. 1896) 230 2.3.5.1 Development of Multiple Regression.

Galton''s Law of Ancestral Heredity 230 2.3.5.2 First Explicit Derivation of the Multivariate Normal Distribution 233 2.3.6 Marriage of Regression with Least Squares (Yule 1897) 237 2.3.7 Correlation Coefficient for a 2 × 2 Table (Yule 1900).

Feud Between Pearson and Yule 244 2.3.8 Intraclass Correlation (Pearson 1901; Harris 1913; Fisher 1921; 1925) 253 2.3.9 First Derivation of the Exact Distribution of r (Fisher 1915) 258 2.3.10 Controversy between Pearson and Fisher on the Latter''s Alleged Use of Inverse Probability (Soper et al. 1917; Fisher 1921) 264 2.

3.11 The Logarithmic (or Z?]) Transformation (Fisher 1915; 1921) 267 *2.3.12 Derivation of the Logarithmic Transformation 270 2.4 Work on Correlation and the Bivariate (and Multivariate) Normal Distribution Before Galton 270 2.4.1 Lagrange''s Derivation of the Multivariate Normal Distribution from the Multinomial Distribution (1776) 271 2.4.

2 Adrain''s Use of the Multivariate Normal Distribution (1808) 275 2.4.3 Gauss'' Use of the Multivariate Normal Distribution in the Theoria Motus (1809) 275 2.4.4 Laplace''s Derivation of the Joint Distribution of Linear Combinations of Two Errors (1811) 276 2.4.5 Plana on the Joint Distribution of Two Linear Combinations of Random Variables (1813) 276 2.4.

6 Bravais'' Determination of Errors in Coordinates (1846) 281 2.4.7 Bullet Shots on a Target: Bertrand''s Derivation of the Bivariate Normal Distribution (1888) 288 3 Karl Pearson''s Chi?]Squared Goodness?]of?]Fit Test 293 3.1 Karl Pearson (1857-1936) 293 3.2 Origin of Pearson''s Chi?]Squared 297 3.2.1 Pearson''s Work on Goodness of Fit Before 1900 297 3.2.

2 Pearson''s 1900 Paper 299 3.3 Pearson''s Error and Clash with Fisher 306 3.3.1 Error by Pearson on the Chi-Squared When Parameters Are Estimated (1900) 306 3.3.2 Greenwood and Yule''s Observation (1915) 308 3.3.3 Fisher''s 1922 Proof of the Chi?]Squared Distribution: Origin of Degrees of Freedom 311 *3.

3.4 Further Details on Degrees of Freedom 313 3.3.5 Reaction to Fisher''s 1922 Paper: Yule (1922), Bowley and Connor (1923), Brownlee (1924), and Pearson (1922) 314 3.3.6 Fisher''s 1924 Argument: "Coup de Gr'ce" in 1926 315 3.3.6.

1 The 1924 Argument 315 3.3.6.2 ''Coup de Gr'ce'' in 1926 317 3.4 The Chi?]Squared Distribution Before Pearson 318 3.4.1 Bienaymé''s Derivation of Simultaneous Confidence Regions (1852) 318 3.4.

2 Abb.