# Mathematical Theory of Probability and Statistics

## 1st Edition

**Author:**Richard von Mises

**Editor:**Hilda Geiringer

**eBook ISBN:**9781483264028

**Imprint:**Academic Press

**Published Date:**1st January 1964

**Page Count:**708

## Description

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)

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 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)

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 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

Inversion

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

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 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)

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 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)

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 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

Tables

Index

## Details

- No. of pages:
- 708

- Language:
- English

- Copyright:
- © Academic Press 1964

- Published:
- 1st January 1964

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483264028

## About the Author

### Richard von Mises

### Affiliations and Expertise

Harvard University Cambridge, Massachusetts