Probability via Expectation

This book has exerted a continuing appeal since its original publication in 1970. It develops the theory of probability from axioms on the expectation functional rather than on probability measure, demonstrates that the standard theory unrolls more naturally and economically this way, and that applications of real interest can be addressed almost immediately. A secondary aim of the original text was to introduce fresh examples and convincing applications, and that aim is continued in this edition, a general revision plus the addition of chapters giving an economical introduction to dynamic programming, that is then applied to the allocation problems represented by portfolio selection and the multi-armed bandit. The investment theme is continued with a critical investigation of the concept of risk-free'trading and the associated Black-Sholes formula, while another new chapter develops the basic ideas of large deviations. The book may be seen as an introduction to probability for students with a basic mathematical facility, covering the standard material, but different in that it is unified by its theme and covers an unusual range of modern applications.

Inhalt

1 Uncertainty, Intuition, and Expectation.- 1 Ideas and Examples.- 2 The Empirical Basis.- 3 Averages over a Finite Population.- 4 Repeated Sampling: Expectation.- 5 More on Sample Spaces and Variables.- 6 Ideal and Actual Experiments: Observables.- 2 Expectation.- 1 Random Variables.- 2 Axioms for the Expectation Operator.- 3 Events: Probability.- 4 Some Examples of an Expectation.- 5 Moments.- 6 Applications: Optimization Problems.- 7 Equiprobable Outcomes: Sample Surveys.- 8 Applications: Least Square Estimation of Random Variables.- 9 Some Implications of the Axioms.- 3 Probability.- 1 Events, Sets and Indicators.- 2 Probability Measure.- 3 Expectation as a Probability Integral.- 4 Some History.- 5 Subjective Probability.- 4 Some Basic Models.- 1 A Model of Spatial Distribution.- 2 The Multinomial, Binomial, Poisson and Geometric Distributions.- 3 Independence.- 4 Probability Generating Functions.- 5 The St. Petersburg Paradox.- 6 Matching, and Other Combinatorial Problems.- 7 Conditioning.- 8 Variables on the Continuum: The Exponential and Gamma Distributions.- 5 Conditioning.- 1 Conditional Expectation.- 2 Conditional Probability.- 3 A Conditional Expectation as a Random Variable.- 4 Conditioning on a ? Field.- 5 Independence.- 6 Statistical Decision Theory.- 7 Information Transmission.- 8 Acceptance Sampling.- 6 Applications of the Independence Concept.- 1 Renewal Processes.- 2 Recurrent Events: Regeneration Points.- 3 A Result in Statistical Mechanics: The Gibbs Distribution.- 4 Branching Processes.- 7 The Two Basic Limit Theorems.- 1 Convergence in Distribution (Weak Convergence).- 2 Properties of the Characteristic Function.- 3 The Law of Large Numbers.- 4 Normal Convergence (the Central Limit Theorem).- 5 The Normal Distribution.- 6 The Law of Large Numbers and the Evaluation of Channel Capacity.- 8 Continuous Random Variables and Their Transformations.- 1 Distributions with a Density.- 2 Functions of Random Variables.- 3 Conditional Densities.- 9 Markov Processes in Discrete Time.- 1 Stochastic Processes and the Markov Property.- 2 The Case of a Discrete State Space: The Kolmogorov Equations.- 3 Some Examples: Ruin, Survival and Runs.- 4 Birth and Death Processes: Detailed Balance.- 5 Some Examples We Should Like to Defer.- 6 Random Walks, Random Stopping and Ruin.- 7 Auguries of Martingales.- 8 Recurrence and Equilibrium.- 9 Recurrence and Dimension.- 10 Markov Processes in Continuous Time.- 1 The Markov Property in Continuous Time.- 2 The Case of a Discrete State Space.- 3 The Poisson Process.- 4 Birth and Death Processes.- 5 Processes on Nondiscrete State Spaces.- 6 The Filing Problem.- 7 Some Continuous-Time Martingales.- 8 Stationarity and Reversibility.- 9 The Ehrenfest Model.- 10 Processes of Independent Increments.- 11 Brownian Motion: Diffusion Processes.- 12 First Passage and Recurrence for Brownian Motion.- 11 Action Optimisation; Dynamic Programming.- 1 Action Optimisation.- 2 Optimisation over Time: the Dynamic Programming Equation.- 3 State Structure.- 4 Optimal Control Under LQG Assumptions.- 5 Minimal-Length Coding.- 6 Discounting.- 7 Continuous-Time Versions and Infinite-Horizon Limits.- 8 Policy Improvement.- 12 Optimal Resource Allocation.- 1 Portfolio Selection in Discrete Time.- 2 Portfolio Selection in Continuous Time.- 3 Multi-Armed Bandits and the Gittins Index.- 4 Open Processes.- 5 Tax Problems.- 13 Finance: 'Risk-Free' Trading and Option Pricing.- 1 Options and Hedging Strategies.- 2 Optimal Targeting of the Contract.- 3 An Example.- 4 A Continuous-Time Model.- 5 How Should it Be Done?.- 14 Second-Order Theory.- 1 Back to L2.- 2 Linear Least Square Approximation.- 3 Projection: Innovation.- 4 The Gauss-Markov Theorem.- 5 The Convergence of Linear Least Square Estimates.- 6 Direct and Mutual Mean Square Convergence.- 7 Conditional Expectations as Least Square Estimates: Martingale Convergence.- 15 Consistency and Extension: The Finite-Dimensional Case.- 1 The Issues.- 2 Convex Sets.- 3 The Consistency Condition for Expectation Values.- 4 The Extension of Expectation Values.- 5 Examples of Extension.- 6 Dependence Information: Chernoff Bounds.- 16 Stochastic Convergence.- 1 The Characterization of Convergence.- 2 Types of Convergence.- 3 Some Consequences.- 4 Convergence in rth Mean.- 17 Martingales.- 1 The Martingale Property.- 2 Kolmogorov's Inequality: the Law of Large Numbers.- 3 Martingale Convergence: Applications.- 4 The Optional Stopping Theorem.- 5 Examples of Stopped Martingales.- 18 Large-Deviation Theory.- 1 The Large-Deviation Property.- 2 Some Preliminaries.- 3 Cramer's Theorem.- 4 Some Special Cases.- 5 Circuit-Switched Networks and Boltzmarm Statistics.- 6 Multi-Class Traffic and Effective Bandwidth.- 7 Birth and Death Processes.- 19 Extension: Examples of the Infinite-Dimensional Case.- 1 Generalities on the Infinite-Dimensional Case.- 2 Fields and ?-Fields of Events.- 3 Extension on a Linear Lattice.- 4 Integrable Functions of a Scalar Random Variable.- 5 Expectations Derivable from the Characteristic Function: Weak Convergence324.- 20 Quantum Mechanics.- 1 The Static Case.- 2 The Dynamic Case.- References.