Author | Whittle, Peter. author |
---|---|

Title | Probability via Expectation [electronic resource] / by Peter Whittle |

Imprint | New York, NY : Springer New York, 1992 |

Edition | Third Edition |

Connect to | http://dx.doi.org/10.1007/978-1-4612-2892-9 |

Descript | XVIII, 300 p. online resource |

SUMMARY

This book is a complete revision of the earlier work Probability which apยญ peared in 1970. While revised so radically and incorporating so much new material as to amount to a new text, it preserves both the aim and the approach of the original. That aim was stated as the provision of a 'first text in probability, deยญ manding a reasonable but not extensive knowledge of mathematics, and taking the reader to what one might describe as a good intermediate level'. In doing so it attempted to break away from stereotyped applications, and consider applications of a more novel and significant character. The particular novelty of the approach was that expectation was taken as the prime concept, and the concept of expectation axiomatized rather than that of a probability measure. In the preface to the original text of 1970 (reproduced below, together with that to the Russian edition of 1982) I listed what I saw as the advantages of the approach in as unlaboured a fashion as I could. I also took the view that the text rather than the author should persuade, and left the text to speak for itself. It has, indeed, stimulated a steady interest, to the point that Springer-Verlag has now commissioned this complete reworking

CONTENT

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 Normal Distribution -- 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 Gaussโ{128}{148}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 in rth Mean -- 14 Martingales -- 1. The Martingale Property -- 2. Kolmogorovโ{128}{153}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

Mathematics
Probabilities
Statistics
Mathematics
Probability Theory and Stochastic Processes
Statistics general