Probability via Expectation
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1 Peter Whittle Probability via Expectation Fourth Edition With 22 Illustrations Springer
2 Contents Preface to the Fourth Edition Preface to the Third Edition Preface to the Russian Edition of Probability (1982) Preface to Probability (1970) vii ix xiii xv 1 Uncertainty, Intuition, and Expectation 1 1 Ideas and Examples 1 2 The Empirical Basis 3 3 Averages over a Finite Population 5 4 Repeated Sampling: Expectation 8 5 More on Sample Spaces and Variables 10 6 Ideal and Actual Experiments: Observables 11 2 Expectation 13 1 Random Variables 13 2 Axioms for the Expectation Operator 14 3 Events: Probability 17 4 Some Examples of an Expectation 18 5 Moments 21 6 Applications: Optimization Problems 22 7 Equiprobable Outcomes: Sample Surveys 24 8 Applications: Least Square Estimation of Random Variables Some Implications of the Axioms 32
3 xviii Contents 3 Probability 39 1 Events, Sets and Indicators 39 2 Probability Measure 43 3 Expectation as a Probability Integral 46 4 SomeHistory 47 5 Subjective Probability 49 4 Some Basic Models 51 1 A Model ofspatial Distribution 51 2 The Multinonüal, Binomial, Poisson and Geometrie Distributions 54 3 Independence 58 4 Probability Generating Functions 61 5 The St. Petersburg Paradox 66 6 Matching, and Other Combinatorial Problems 68 7 Conditioning 71 8 Variables on the Continuum: The Exponential and Gamma Distributions 76 5 Conditioning 80 1 Conditional Expectation 80 2 Conditional Probability 84 3 A Conditional Expectation as a Random Variable 88 4 Conditioning ona er-field 92 5 Independence 93 6 Statistical Decision Theory 95 7 Information Transmission 97 8 Acceptance Sampling 99 6 Applications of the Independence Concept Renewal Processes Recurrent Events: Regeneration Points A Result in Statistical Mechanics: The Gibbs Distribution Brandung Processes The Two Basic Limit Theorems Convergence in Distribution (Weak Convergence) PropertiesoftheCharacteristic Function TheLawofLargeNumbers Normal Convergence (the Central Limit Theorem) The Normal Distribution The Law of Large Numbers and the Evaluation of Channel Capacity 138 r\
4 Contents xix 8 Continuous Random Variables and Their Transformations Distributions with a Density Functions of Random Variables Conditional Densities Markov Processes in Discrete Time Stochastic Processes and the Markov Property The Case of a Discrete State Space: The Kolmogorov Equations Some Examples: Ruin, Survival and Runs Birth and Death Processes: Detailed Balance Some Examples We Should Like to Defer Random Walks, Random Stopping and Ruin Auguries of Martingales Recurrence and Equilibrium Recurrence and Dimension Markov Processes in Continuous Time The Markov Property in Continuous Time The Case ofa Discrete State Space The Poisson Process Birth and Death Processes Processes on Nondiscrete State Spaces The Filing Problem Some Continuous-Time Martingales Stationarity and Reversibility The Ehrenfest Model Processes of Independent Increments Brownian Motion: Diffusion Processes First Passage and Recurrence for Brownian Motion Action Optimisation; Dynamic Programming Action Optimisation Optimisation over Time: the Dynamic Programming Equation State Structure Optimal Control Under LQG Assumptions Minimal-Length Coding Discounting Continuous-Time Versions and Infinite-Horizon Limits Policy Improvement Optimal Resource Allocation Portfolio Selection in Discrete Time Portfolio Selection in Continuous Time 232
5 xx Contents 3 Multi-Armed Bandits and the Gittins Index Open Processes Tax Problems Finance: 'Risk-Free' Trading and Option Pricing Options and Hedging Strategies Optimal Targeting of the Contract An Example A Continuous-Time Model HowS/ioj/WitBeDone? Second-Order Theory Backte L Linear Least Square Approximation Projection: Innovation The Gauss-Markov Theorem The Convergence of Linear Least Square Estimates Direct and Mutual Mean Square Convergence Conditional Expectations as Least Square Estimates: Martingale Convergence Consistency and Extension: The Finite-Dimensional Case Thelssues ConvexSets The Consistency Condition for Expectation Values The Extension of Expectation Values Examples of Extension Dependence Information: Chernoff Bounds Stochastic Convergence The Characterization of Convergence Types of Convergence Some Consequences Convergence in rth Mean Martingales The Martingale Property Kolmogorov's Inequality: the Law of Large Numbers Martingale Convergence: Applications The Optional Stopping Theorem Examples of Stopped Martingales...., Large-Deviation Theory The Large-Deviation Property Some Preliminaries Cramer's Theorem 309
6 Contents xxi 4 Some Special Cases Circuit-Switched Networks and Boltzmann Staüstics Multi-Class Traffic and Effective Bandwidth Birth and Death Processes Extension: Examples of the Infinite-Dimensional Case Generalities on the Infinite-Dimensional Case Fields and or-fieldsof Events Extension on a Linear Lattice Integrable Functions of a Scalar Random Variable Expectations Derivable from the Characteristic Function: Weak Convergence Quantum Mechanics The Static Case The Dynamic Case 335 References 341 Index 345
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