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# Von Mises R., Geiringer H. Mathematical Theory of Probability and Statistics

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New York: Elsevier, 1964. — 698 p.
Mathematical Theory of Probability and Statistics focuses on the contributions and influence of Richard von Mises on the processes, methodologies, and approaches involved in the mathematical theory of probability and statistics.
The publication first elaborates on fundamentals, general label space, and basic properties of distributions. Discussions focus on Gaussian distribution, Poisson distribution, mean value variance and other moments, non-countable label space, basic assumptions, operations, and distribution function. The text then ponders on examples of combined operations and summation of chance variables characteristic function.
The book takes a look at the asymptotic distribution of the sum of chance variables and probability inference. Topics include inference from a finite number of observations, law of large numbers, asymptotic distributions, limit distribution of the sum of independent discrete random variables, probability of the sum of rare events, and probability density. The text also focuses on the introduction to the theory of statistical functions and multivariate statistics.
Contents:
Preface
Fundamentals
The Basic Assumptions
Introduction
Sequences of Observations. The Label Space
Frequency. Chance
Randomness
The Collective
The Operations
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 Results
Appendix Two: Measure-Theoretical Approach versus Frequency Approach
General Label Space
Distribution Function (Discrete Case). Measure-Theoretical Approach
Introduction
Cumulative Distribution Function for the Discrete Case
Non-Countable Label Space. Measure-Theoretical Approach
Non-Countable Label Space. Frequency Approach
The Field of Definition of Probability in a Frequency Theory
Basic Extension
The Field Fx
Distribution Function. Riemann-Stieltjes Integral. Probability Density
Appendix Three: Tornier's Frequency Theory
Basic Properties of Distributions
Mean Value, Variance, and Other Moments
Mean Value and Variance. Tchebycheff's Inequality
Expectation Relative to a Distribution. Stieltjes Integral
Generalizations of Tchebycheff's Inequality
Moments of a Distribution
Gaussian Distribution, Poisson Distribution
The Normal or Gaussian Distribution in One Dimension
The Poisson Distribution
Distributions in Rn
Distributions in More Than One Dimension
Mean Value and Variance in Several Dimensions
Examples of Combined Operations
Uniform Distributions
Uniform Arithmetical Distribution
Uniform Density. Needle Problem
Bernoulli Problem and Related Questions
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
Some Problems of Non-independent Events
A Problem of Runs
Application to Mendelian Heredity Theory
Basic Facts and Definitions
Definitions. Classification
Applications of Markov Chains
Summation of Chance Variables Characteristics Function
Summation of Chance Variables and Laws of Large Numbers
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
Characteristic Function
The Characteristic Function
Inversion
Solution of the Summation Problem. Stability of the Normal Distribution and of the Poisson Distribution
Continuity Theorem for Characteristic Functions
Asymptotic Distribution of the Sum of Chance Variables
Asymptotic Results for Infinite Products. Stirling's Formula. Laplace's Formula
Product of an Infinite Number of Functions
Application of the Product Formulas
Limit Distribution of the Sum of Independent Discrete Random Variables
Arithmetical Probabilities
Examples
Probability Density. Central Limit Theorem. Lindeberg's and Liapounoff's Conditions
The Summation Problem in the General Case
The Central Limit Theorem. Necessary and Sufficient Conditions
Liapounoff's Sufficient Condition
Probability of the Sum of Rare Events. Compound Poisson Distribution
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 8
Appendix Four : Remarks on Additive Time-Dependent Stochastic Processes
Probability Inference. Bayes' Method
Inference from a Finite Number of Observations
Bayes' Problem and Solution
Discussion of p0(x). Assumption P0(x) = constant
Law of Large Numbers
Bayes' Theorem. Irrelevance of p0(x) for large n
Asymptotic Distributions
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
Rare Events
Inference on the Probability of Rare Events
More on Distributions
Sample Distribution and Statistical Parameters
Repartition
Some Statistical Parameters
Expectations and Variances of Sample Mean and Sample Variance
Moments. Inequalities
Determining a Distribution by Its First (2m — 1) Moments
Some Inequalities
Various Distributions Related to Normal Distributions
The Chi-Square Distribution. Some Applications
Student's Distribution and F Distribution
Multivariate Normal Distribution
Normal Distribution in Three Dimensions
Properties of the Multivariate Normal Distribution
Analysis of Statistical Data
Lexis Theory
Repeated Equal Alternatives
Non-Equal Alternatives
Student Test and F-Test
The Two Tests
The X2-Test
Checking a Known Distribution
X2-Test if Certain Parameters of the Theoretical Distribution Are Estimated from the Sample
The w2-Tests
von Mises' Definition
Smirnov's w2-Test
Deviation Tests
Problem of Inference
Testing Hypotheses
The Basis of Statistical Inference
Testing Hypotheses. Introduction of Neyman-Pearson Method
Neyman-Pearson Method. Composite Hypothesis. Discontinuous and Multivariate Cases
On Sequential Sampling
Global Statements on Parameters
Confidence Limits
Estimation
Maximum Likelihood Method
Further Remarks on Estimation
Multivariate Statistics. Correlation
Measures of Correlation in Two Dimensions
Correlation
Regression Lines
Other Measures of Correlation
Distribution of the Correlation Coefficient
Asymptotic Expectation and Variance of the Correlation Coefficient
The Distribution of r in Normal Samples
Generalizations to k Variables
Regression and Correlation in k Variables
Remarks on the Distribution of Correlation Measures from a k-Dimensional Normal Population
Asymptotic Expectation and Variance of Statistical Functions
Introduction to the Theory of Statistical Functions
Differentiable Statistical Functions
Statistical Functions. Continuity, Differentiability
Higher Derivatives. Taylor's Theorem
The Laws of Large Numbers
The First Law of Large Numbers for Statistical Functions
The Second Law of Large Numbers for Statistical Functions
Statistical Functions of Type One
Convergence Toward the Normal Distribution
Convergence toward the Gaussian Distribution. General Case
Classification of Differentiable Statistical Functions
Asymptotic Expressions for Expectations
Asymptotic Behavior of Statistical Functions
Selected Reference Books
Tables
Index
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