Probability via Expectation

This classic text, now in its third edition, has been widely used as an introduction to probability. Its main aim is to present a straightforward introduction to the main concepts and applications of probability at an undergraduate level. Historically, the early analysts of games of chance found the question 'What is the fair price for entering this game?' as natural a question as 'What is the probability of winning it?'. This book differs from many textbooks in that the author takes as the starting point for the subject's development expectation rather than the traditional probability measure approach. All the main concepts of a first course in probability are covered including probability measures, independence, conditional probability, the basic limit theorems, and Markov processes. Throughout, the author stresses the importance of applications and includes numerous examples covering a range of difficulties. Little is required in the way of prerequisites - a basic exposure to calculus and matrix algebra will be sufficient for any student to enjoy this first course in probability.

List 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.

Product details

• Authors: P. Whittle, Peter Whittle
• Assisted by: Stephen E. Fienberg (Editor), Ingram Olkin (Editor)
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.1992
• EAN: 9780387977645
• ISBN: 978-0-387-97764-5
• No. of pages: 300
• Weight: 450 g
• Illustrations: Illustr.
• Series: Springer Texts in Statistics; Springer Texts in Statistics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics