Mathematical Theory of Probability and Statistics

PrefaceChapter I Fundamentals A. The Basic Assumptions (Sections 1-5) Introduction Sequences of Observations. The Label Space Frequency. Chance Randomness The Collective B. The Operations (Sections 6-10) First Operation: Place Selection Second Operation: Mixing. Probability as Measure Third Operation: Partition Fourth Operation: Combining Additional Remarks on IndependenceAppendix One: The Consistency of the Notion of the Collective. Wald's ResultsAppendix Two: Measure-Theoretical Approach versus Frequency ApproachChapter II General Label Space A. Distribution Function (Discrete Case). Measure-Theoretical Approach (Sections 1-3) Introduction Cumulative Distribution Function for the Discrete Case Non-Countable Label Space. Measure-Theoretical Approach B. Non-Countable Label Space. Frequency Approach (Sections 4-7) The Field of Definition of Probability in a Frequency Theory Basic Extension The Field Fx Distribution Function. Riemann-Stieltjes Integral. Probability DensityAppendix Three: Tornier's Frequency TheoryChapter III Basic Properties of Distributions A. Mean Value, Variance, and Other Moments (Sections 1-4) Mean Value and Variance. Tchebycheff's Inequality Expectation Relative to a Distribution. Stieltjes Integral Generalizations of Tchebycheff's Inequality Moments of a Distribution B. Gaussian Distribution, Poisson Distribution (Sections 5 and 6) The Normal or Gaussian Distribution in One Dimension The Poisson Distribution C. Distributions in Rn (Sections 7 and 8) Distributions in More Than One Dimension Mean Value and Variance in Several DimensionsChapter IV Examples of Combined Operations A. Uniform Distributions (Sections 1 and 2) Uniform Arithmetical Distribution Uniform Density. Needle Problem B. Bernoulli Problem and Related Questions (Sections 3-6) The Problem of Repeated Trials Bernoulli's Theorem The Approximation to the Binomial Distribution in the Case of Rare Events. Poisson Distribution The Negative Binomial Distribution C. Some Problems of Non-independent Events (Sections 7-9) A Problem of Runs Arbitrarily Linked Events. Basic Relations Examples of Arbitrarily Linked Events D. Application to Mendelian Heredity Theory (Sections 10 and 11) Basic Facts and Definitions Probability Theory of Linkage E. Comments On Markov Chains (Sections 12 and 13) Definitions. Classification Applications of Markov ChainsChapter V Summation of Chance Variables Characteristics Function A. Summation of Chance Variables and Laws of Large Numbers (Sections 1-4) Summation of Chance Variables The Laws of Large Numbers Laws of Large Numbers Continued. Khintchine's Theorem. Markov's Theorem Strong Laws of Large Numbers B. Characteristic Function (Sections 5-8) The Characteristic Function Inversion Solution of the Summation Problem. Stability of the Normal Distribution and of the Poisson Distribution Continuity Theorem for Characteristic FunctionsChapter VI Asymptotic Distribution of the Sum of Chance Variables A. Asymptotic Results for Infinite Products. Stirling's Formula. Laplace's Formula (Sections 1 and 2) Product of an Infinite Number of Functions Application of the Product Formulas B. Limit Distribution of the Sum of Independent Discrete Random Variables (Sections 3 and 4) Arithmetical Probabilities Examples C. Probability Density. Central Limit Theorem. Lindeberg's and Liapounoff's Conditions (Sections 5-7) The Summation Problem in the General Case The Central Limit Theorem. Necessary and Sufficient Conditions Liapounoff's Sufficient Condition D. Probability of the Sum of Rare Events. Compound Poisson Distribution (Sections 8-10) Asymptotic Distribution of the Sum of n Discrete Random Variables in the Case of Rare Events Limit Probability of the Sum of Rare Events as a Compound Poisson Distribution A Generalization of the Theorem of Section 8Appendix Four : Remarks on Additive Time-Dependent Stochastic ProcessesChapter VII Probability Inference. Bayes' Method A. Inference from a Finite Number of Observations (Sections 1 and 2) Bayes' Problem and Solution Discussion of p0(x). Assumption P0(x) = constant B. Law of Large Numbers (Section 3) Bayes' Theorem. Irrelevance of p0(x) for large n C. Asymptotic Distributions (Sections 4-6) Limit Theorems for Bayes' Problem Application of the Two Basic Limit Theorems to the Theory of Errors Inference on a Statistical Function of Unknown Probabilities D. Rare Events (Section 7) Inference on the Probability of Rare EventsChapter VIII More on Distributions A. Sample Distribution and Statistical Parameters (Section 1-3) Repartition Some Statistical Parameters Expectations and Variances of Sample Mean and Sample Variance B. Moments. Inequalities (Sections 4 and 5) Determining a Distribution by Its First (2m — 1) Moments Some Inequalities C. Various Distributions Related to Normal Distributions (Sections 6 and 7) The Chi-Square Distribution. Some Applications Student's Distribution and F Distribution D. Multivariate Normal Distribution (Sections 8 and 9) Normal Distribution in Three Dimensions Properties of the Multivariate Normal DistributionChapter IX Analysis of Statistical Data A. Lexis Theory (Sections 1 and 2) Repeated Equal Alternatives Non-Equal Alternatives B. Student Test and F-Test (Section 3) The Two Tests C. The X2-Test (Sections 4 and 5) Checking a Known Distribution X2-Test if Certain Parameters of the Theoretical Distribution Are Estimated from the Sample D. The w2-Tests (Sections 6 and 7) von Mises' Definition Smirnov's w2-Test E. Deviation Tests (Section 8) On the Kolmogorov-Smirnov TestsChapter X Problem of Inference A. Testing Hypotheses (Sections 1-4) The Basis of Statistical Inference Testing Hypotheses. Introduction of Neyman-Pearson Method Neyman-Pearson Method. Composite Hypothesis. Discontinuous and Multivariate Cases On Sequential Sampling B. Global Statements on Parameters (Section 5) Confidence Limits C. Estimation (Sections 6 and 7) Maximum Likelihood Method Further Remarks on EstimationChapter XI Multivariate Statistics. Correlation A. Measures of Correlation in Two Dimensions (Sections 1-3) Correlation Regression Lines Other Measures of Correlation B. Distribution of the Correlation Coefficient (Sections 4 and 5) Asymptotic Expectation and Variance of the Correlation Coefficient The Distribution of r in Normal Samples C. Generalizations to k Variables (Sections 6 and 7) Regression and Correlation in k Variables Remarks on the Distribution of Correlation Measures from a k-Dimensional Normal Population D. First Comments on Statistical Functions (Section 8) Asymptotic Expectation and Variance of Statistical FunctionsChapter XII Introduction to the Theory of Statistical Functions A. Differentiable Statistical Functions (Sections 1 and 2) Statistical Functions. Continuity, Differentiability Higher Derivatives. Taylor's Theorem B. The Laws of Large Numbers (Sections 3 and 4) The First Law of Large Numbers for Statistical Functions The Second Law of Large Numbers for Statistical Functions C. Statistical Functions of Type One (Sections 5 and 6) Convergence Toward the Normal Distribution Convergence toward the Gaussian Distribution. General Case D. Classification of Differentiable Statistical Functions (Sections 7 and 8) Asymptotic Expressions for Expectations Asymptotic Behavior of Statistical FunctionsSelected Reference BooksTablesIndex