Mathematical Theory of Probability and Statistics
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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)
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
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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