Agnes Scott College

Richard von Mises, Edited and Complemented by Hilda Geiringer

Mathematical Theory of Probability and Statistics
Academic Press, New York and London, 1964

Preface Excerpts (by Hilda Geiringer)

Richard von Mises' scientific work comprises two major fields: mechanics and probability-statistics; supplemented by numerical analysis, geometry, and philosophy of science they form the leading interests of his scientific life.

In 1931 he published a comprehensive textbook on probability consisting of four parts: the foundations (his frequency theory); limit theorems; statistics and theory of errors; and statistical problems in physics. While in the United States (1939-1953) he became deeply interested in "British-American" statistics (as did his former student A. Wald). Mises' aim was to understand this approach to statistical inference as a part of rigorous probability theory, its application. The results of his thinking were incorporated in Lectures on Probability and Statistics he gave repeatedly at Harvard University to advanced undergraduate and graduate students, and in lectures he gave in Rome (1951-1952) and finally in Zurich(summer 1952).

The Harvard Lectures were mimeographed. Brief but clear notes of the Zurich Lectures were kindly given to me after Mises' death by K. Schoeni, who attended them. Mises had planed to incorporate the various ideas in a comprehensive work on probability and statistics which, more than 20 years after his Wahrscheinlickkeitscrechnung, would have been a very different work, in many respects.

The present book is based on the material mentioned above as well as his papers and notebooks. It presents a unified mathematical theory of probability and statistics. In fact, for Mises there were never two different theories, one "pure" the other "applied," but one theory only, a frequency theory, mathematically rigorous and guided by an operational approach.

Contents

Chapter 1. Fundamentals

  1. The Basic Assumptions (Sections 1-5)
  2. The Operations (Sections 6-10)

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

Appendix Two: Measure-Theoretic Approach versus Frequency Approach

Chapter II. General Label Space

  1. Distribution Function (Discrete Case). Measure-Theoretic Approach (Sections 1-3)
  2. Non-Countable Label Space. Frequency Approach (Sections 4-7)

Appendix Three: Tornier's Frequency Theory

Chapter III. Basic Properties of Distributions

  1. Mean Value, Variance, and Other Moments (Sections 1-4)
  2. Gaussian Distribution, Poisson Distribution (Sections 5 and 6)
  3. Distributions in Rn (Sections 7 and 8)

Chapter IV. Examples of Combined Operations

  1. Uniform Distributions (Sections 1 and 2)
  2. Bernoulli Problem and Related Questions (Sections 3-6)
  3. Some Problems of Non-Independent Events (Sections 7-9)
  4. Application to Mendelian Heredity Theory (Sections 10 and 11)
  5. Comments on Markov Chains (Sections 12 and 13)

Chapter V. Summation of Chance Variables Characteristic Function

  1. Summation of Chance Variables and Laws of Large Numbers (Sections 1-4)
  2. Characteristic Function (Sections 5-8)

Chapter VI. Asymptotic Distribution of the Sum of Chance Variables

  1. Asymptotic Results for Infinite Products. Stirling's Formula. Laplace's Formula (Sections 1 and 2)
  2. Limit Distribution of the Sum of Independent Discrete Random Variables (Sections 3 and 4)
  3. Probability Density. Central Limit Theorem. Lindeberg's and Liapounoff's Conditions (Sections 5-7)
  4. Probability of the Sum of Rare Events. Compound Poisson Distribution (Sections 8-10)

Appendix Four: Remarks on Additive Time-Dependent Stochastic Processes

Chapter VII. Probability Inference. Bayes' Method

  1. Inference from a Finite Number of Observations (Sections 1 and 2)
  2. Law of Large Numbers (Section 3)
  3. Asymptotic Distributions (Sections 4-6)
  4. Rare Events (Section 7)

Chapter VIII. More on Distributions

  1. Sample Distribution and Statistical Parameters (Sections 1-3)
  2. Moments. Inequalities (Sections 4 and 5)
  3. Various Distributions Related to Normal Distributions (Sections 6 and 7)
  4. Multivariate Normal Distribution (Sections 8 and 9)

Chapter IX. Analysis of Statistical Data

  1. Lexis Theory (Sections 1 and 2)
  2. Student Test and F-Test (Section 3)
  3. The X2-Test (Sections 4 and 5)
  4. The ω2-Tests (Sections 6 and 7)
  5. Deviation Tests (Section 8)

Chapter X. Problem of Inference

  1. Testing Hypotheses (Sections 1-4)
  2. Global Statements on Parameters (Section 5)
  3. Estimation (Sections 6 and 7)

Chapter XI. Multivariate Statistics. Correlation

  1. Measures of Correlation in Two Dimensions (Sections 1-3)
  2. Distribution of the Correlation Coefficient (Sections 4 and 5)
  3. Generalizations to k Variables (Sections 6 and 7)
  4. First Comments on Statistical Functions (Section 8)

Chapter XII. Introduction to the Theory of Statistical Functions

  1. Differentiable Statistical Functions (Sections 1 and 2)
  2. The Laws of Large Numbers (Sections 3 and 4)
  3. Statistical Functions of Type One (Sections 5 and 6)
  4. Classification of Differentiable Statistical Functions (Sections 7 and 8)