Table of Contents

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 ?-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 NormalDistribution.- 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 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.- 12 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.- 13 Stochastic Convergence.- 1. The Characterization of Convergence.- 2. Types of Convergence.- 3. Some Consequences.- 4. Convergence inrth Mean.- 14 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.- 15 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 Convergence.- 16 Some Interesting Processes.- 1. Information Theory: Block Coding.- 2. Information Theory: More on the Shannon Measure.- 3. Information Theory: Sequential Interrogation and Questionnaires.- 4. Dynamic Optimization.- 5. Quantum Mechanics: the Static Case.- 6. Quantum Mechanics: the Dynamic Case.- References.