Peter Whittle

Probability via Expectation (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 353
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461205098
• Artikelnr.: 44058543

Inhaltsangabe

Uncertainty, Intuition and Expectation.- Expectation.- Probability.- Some Basic Models.- Conditioning.- Applications of the Independence Concept.- The Two Basic Limit Theorems.- Continuous Random Variables and Their Transformations.- Markov Processes in Discrete Time.- Markov Processes in Continuous Time.- Action Optimisation: Dynamic Programming.- Optimal Resource Allocation.- Finance: Option Pricing and the Implied Martingale.- Second-Order Theory.- Consistency and Extension: The Finite-Dimensional Case.- Stochastic Convergence.- Martingales.- Extension: Examples of the Infinite-Dimensional Case.- Large-Deviation Theory.- Quantum Mechanics.

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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.

Rezensionen

... a clear success in its unorthodoxy, Probability via Expectation has become a treasured classic. P.A.L. Emrechts in "Short Book Rezensions", Vol. 21/1, April,.

"This surprising and beautiful introduction to concepts of probability ... chapters have been added which deal with areas of big actual interest ... ." (Peter Imkeller, zbMATH 0980.60004, 2022)

From the reviews of the fourth edition:

"... a clear success in its unorthodoxy, Probability via Expectation has become a treasured classic."
P.A.L. Emrechts in "Short Book Reviews", Vol. 21/1, April, 2001

"Apart from presenting a case for the development of probability theory by using the expectation operator rather than probability measure as the primitive notion, a second distinctive feature of this book is the very large range of modern applications that it covers. Many of these are addressed by more than 350 exercises interspersed throughout the text. In summary, this well written book is a ... introduction to probability theory and its applications." (Norbert Henze, Metrika, November, 2002)

"Originally published in 1970, this book has stood the test of time. ... the text demonstrates a modern alternative approach to a now classical field. ... The fourth edition contains a number of modifications and corrections. New material on dynamic programming, optimal allocation, options pricing and large deviations is included." (Martin T. Wells, Journal of the American Statistical Association, September 2001)