Mathematical Theory of Probability and Statistics

Mathematical Theory of Probability and Statistics

1st Edition - January 1, 1964

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  • Author: Richard von Mises
  • eBook ISBN: 9781483264028

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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. The publication is a dependable source of information for researchers interested in the mathematical theory of probability and statistics

Table of Contents

  • Preface

    Chapter I Fundamentals

    A. The Basic Assumptions (Sections 1-5)


    Sequences of Observations. The Label Space

    Frequency. Chance


    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 Independence

    Appendix One: The Consistency of the Notion of the Collective. Wald's Results

    Appendix Two: Measure-Theoretical Approach versus Frequency Approach

    Chapter II General Label Space

    A. Distribution Function (Discrete Case). Measure-Theoretical Approach (Sections 1-3)


    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 Density

    Appendix Three: Tornier's Frequency Theory

    Chapter 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 Dimensions

    Chapter 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 Chains

    Chapter 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


    Solution of the Summation Problem. Stability of the Normal Distribution and of the Poisson Distribution

    Continuity Theorem for Characteristic Functions

    Chapter 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


    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 8

    Appendix Four : Remarks on Additive Time-Dependent Stochastic Processes

    Chapter 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 Events

    Chapter VIII More on Distributions

    A. Sample Distribution and Statistical Parameters (Section 1-3)


    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 Distribution

    Chapter 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 Tests

    Chapter 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 Estimation

    Chapter XI Multivariate Statistics. Correlation

    A. Measures of Correlation in Two Dimensions (Sections 1-3)


    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 Functions

    Chapter 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 Functions

    Selected Reference Books



Product details

  • No. of pages: 708
  • Language: English
  • Copyright: © Academic Press 1964
  • Published: January 1, 1964
  • Imprint: Academic Press
  • eBook ISBN: 9781483264028

About the Author

Richard von Mises

Affiliations and Expertise

Harvard University Cambridge, Massachusetts

About the Editor

Hilda Geiringer

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