book

Theory of Probability

Name: Theory of Probability
Author: Bruno de Finetti
ISBN: 9781119286370

by Bruno de Finetti

April 2017

Intermediate to advanced

600 pages

25h 37m

English

Wiley

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Cover
Title Page
Foreword
Preface
1 Introduction
1.1 Why a New Book on Probability?1.2 What are the Mathematical Differences?1.3 What are the Conceptual Differences?1.4 Preliminary Clarifications1.5 Some Implications to Note1.6 Implications for the Mathematical Formulation1.7 An Outline of the ‘Introductory Treatment’1.8 A Few Words about the ‘Critical’ Appendix1.9 Other Remarks1.10 Some Remarks on Terminology1.11 The Tyranny of Language1.12 References
2 Concerning Certainty and Uncertainty
2.1 Certainty and Uncertainty2.2 Concerning Probability2.3 The Range of Possibility2.4 Critical Observations Concerning the ‘Space of Alternatives’2.5 Logical and Arithmetic Operations2.6 Assertion, Implication; Incompatibility2.7 Partitions; Constituents; Logical Dependence and Independence2.8 Representations in Linear form2.9 Means; Associative Means2.10 Examples and Clarifications2.11 Concerning Certain Conventions of Notation
3 Prevision and Probability
3.1 From Uncertainty to Prevision3.2 Digressions on Decisions and Utilities3.3 Basic Definitions and Criteria3.4 A Geometric Interpretation: The Set 𝓟 of Coherent Previsions3.5 Extensions of Notation3.6 Remarks and Examples3.7 Prevision in the Case of Linear and Nonlinear Dependence3.8 Probabilities of Events3.9 Linear Dependence in General3.10 The Fundamental Theorem of Probability3.11 Zero Probabilities: Critical Questions3.12 Random Quantities with an Infinite Number of Possible Values3.13 The Continuity Property
4 Conditional Prevision and Probability
4.1 Prevision and the State of Information4.2 Definition of Conditional Prevision (and Probability)4.3 Proof of the Theorem of Compound Probabilities4.4 Remarks4.5 Probability and Prevision Conditional on a Given Event H4.6 Likelihood4.7 Probability Conditional on a Partition H4.8 Comments4.9 Stochastic Dependence and Independence; Correlation4.10 Stochastic Independence Among (Finite) Partitions4.11 On the Meaning of Stochastic Independence4.12 Stochastic Dependence in the Direct Sense4.13 Stochastic Dependence in the Indirect Sense4.14 Stochastic Dependence through an Increase in Information4.15 Conditional Stochastic Independence4.16 Noncorrelation; Correlation (Positive or Negative)4.17 A Geometric Interpretation4.18 On the Comparability of Zero Probabilities4.19 On the Validity of the Conglomerative Property
5 The Evaluation of Probabilities
5.1 How should Probabilities be Evaluated?5.2 Bets and Odds5.3 How to Think about Things5.4 The Approach Through Losses5.5 Applications of the Loss Approach5.6 Subsidiary Criteria for Evaluating Probabilities5.7 Partitions into Equally Probable Events5.8 The Prevision of a Frequency5.9 Frequency and ‘Wisdom after the Event’5.10 Some Warnings5.11 Determinism, Indeterminism, and other ‘Isms’
6 Distributions
6.1 Introductory Remarks6.2 What we Mean by a ‘Distribution’6.3 The Parting of the Ways6.4 Distributions in Probability Theory6.5 An Equivalent Formulation6.6 The Practical Study of Distribution Functions6.7 Limits of Distributions6.8 Various Notions of Convergence for Random Quantities6.9 Distributions in Two (or More) Dimensions6.10 The Method of Characteristic Functions6.11 Some Examples of Characteristic Functions6.12 Some Remarks Concerning the Divisibility of Distributions

7 A Preliminary Survey
7.1 Why a Survey at this Stage?7.2 Heads and Tails: Preliminary Considerations7.3 Heads and Tails: The Random Process7.4 Some Particular Distributions7.5 Laws of ‘Large Numbers’7.6 The ‘Central Limit Theorem’; The Normal Distribution7.7 Proof of the Central Limit Theorem
8 Random Processes with Independent Increments
8.1 Introduction8.2 The General Case: The Case of Asymptotic Normality8.3 The Wiener–Lévy Process8.4 Stable Distributions and Other Important Examples8.5 Behaviour and Asymptotic Behaviour8.6 Ruin Problems; the Probability of Ruin; the Prevision of the Duration of the Game8.7 Ballot Problems; Returns to Equilibrium; Strings8.8 The Clarification of Some So‐Called Paradoxes8.9 Properties of the Wiener–Lévy Process
9 An Introduction to Other Types of Stochastic Process
9.1 Markov Processes9.2 Stationary Processes
10 Problems in Higher Dimensions
10.1 Introduction10.2 Second‐Order Characteristics and the Normal Distribution10.3 Some Particular Distributions: The Discrete Case10.4 Some Particular Distributions: The Continuous Case10.5 The Case of Spherical Symmetry
11 Inductive Reasoning; Statistical Inference
11.1 Introduction11.2 The Basic Formulation and Preliminary Clarifications11.3 The Case of Independence and the Case of Dependence11.4 Exchangeability
12 Mathematical Statistics
12.1 The Scope and Limits of the Treatment12.2 Some Preliminary Remarks12.3 Examples Involving the Normal Distribution12.4 The Likelihood Principle and Sufficient Statistics12.5 A Bayesian Approach to ‘Estimation’ and ‘Hypothesis Testing’12.6 Other Approaches to ‘Estimation’ and ‘Hypothesis Testing’12.7 The Connections with Decision Theory
Appendix
1 Concerning Various Aspects of the Different Approaches2 Events (true, false, and …)3 Events in an Unrestricted Field4 Questions Concerning ‘Possibility’5 Verifiability and the Time Factor6 Verifiability and the Operational Factor7 Verifiability and the Precision Factor8 Continuation: The Higher (or Infinite) Dimensional Case9 Verifiability and ‘Indeterminism’10 Verifiability and ‘Complementarity’11 Some Notions Required for a Study of the Quantum Theory Case12 The Relationship with ‘Three‐Valued Logic’13 Verifiability and Distorting Factors14 From ‘Possibility’ to ‘Probability’15 The First and Second Axioms16 The Third Axiom17 Connections with Aspects of the Interpretations18 Questions Concerning the Mathematical Aspects19 Questions Concerning Qualitative Formulations20 Conclusions
Index
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Content preview from Theory of Probability

10Problems in Higher Dimensions

10.1 Introduction

10.1.1. It might be argued that every problem could, or even should, be put in a multidimensional framework; indeed, we have seen this over and over again throughout our treatment so far. The subject matter of this chapter is not really new, therefore, and we shall merely emphasize those features and problems which particularly relate to the multi‐dimensional nature of certain distributions.

In Chapter 6, 6.9.1, we dealt with the essential points concerning the representation of a distribution over an r‐dimensional Cartesian space, either by means of the distribution function

(10.1)

or, if it exists, by means of the density

(10.2)

In addition, we can state that a necessary and sufficient condition for a function F(x_l, x₂,…, x_r) to be a distribution function is that ƒ never be non‐negative, or, should ƒ not exist, that the expression for which it would be the limit is non‐negative. The latter is the probability of the rectangular prism given by

(10.3)

the sum being taken over the 2^r vertices corresponding to all possible ...

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Theory of Probability

by Bruno de Finetti

10Problems in Higher Dimensions

10.1 Introduction

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