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.