Basic Probability Theory
by Ash, Robert B.Format: Paperback
Pub. Date: 2008-06-26
Publisher(s): Dover Publications
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Summary
This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus.
Author Biography
Robert B. Ash is Professor of Mathematics at the University of Illinois.
Table of Contents
| Basic Concepts | |
| Introduction | p. 1 |
| Algebra of Events (Boolean Algebra) | p. 3 |
| Probability | p. 10 |
| Combinatorial Problems | p. 15 |
| Independence | p. 25 |
| Conditional Probability | p. 33 |
| Some Fallacies in Combinatorial Problems | p. 39 |
| Appendix: Stirling's Formula | p. 43 |
| Random Variables | |
| Introduction | p. 46 |
| Definition of a Random Variable | p. 48 |
| Classification of Random Variables | p. 51 |
| Functions of a Random Variable | p. 58 |
| Properties of Distribution Functions | p. 66 |
| Joint Density Functions | p. 70 |
| Relationship Between Joint and Individual Densities; Independence of Random Variables | p. 76 |
| Functions of More Than One Random Variable | p. 85 |
| Some Discrete Examples | p. 95 |
| Expectation | |
| Introduction | p. 100 |
| Terminology and Examples | p. 107 |
| Properties of Expectation | p. 114 |
| Correlation | p. 119 |
| The Method of Indicators | p. 122 |
| Some Properties of the Normal Distribution | p. 124 |
| Chebyshev's Inequality and the Weak Law of Large Numbers | p. 126 |
| Conditional Probability and Expectation | |
| Introduction | p. 130 |
| Examples | p. 133 |
| Conditional Density Functions | p. 135 |
| Conditional Expectation | p. 140 |
| Appendix: The General Concept of Conditional Expectation | p. 152 |
| Characteristic Functions | |
| Introduction | p. 154 |
| Examples | p. 158 |
| Properties of Characteristic Functions | p. 166 |
| The Central Limit Theorem | p. 169 |
| Infinite Sequences of Random Variables | |
| Introduction | p. 178 |
| The Gambler's Ruin Problem | p. 182 |
| Combinatorial Approach to the Random Walk; the Reflection Principle | p. 186 |
| Generating Functions | p. 191 |
| The Poisson Random Process | p. 196 |
| The Strong Law of Large Numbers | p. 203 |
| Markov Chains | |
| Introduction | p. 211 |
| Stopping Times and the Strong Markov Property | p. 217 |
| Classification of States | p. 220 |
| Limiting Probabilities | p. 230 |
| Stationary and Steady-State Distributions | p. 236 |
| Introduction to Statistics | |
| Statistical Decisions | p. 241 |
| Hypothesis Testing | p. 243 |
| Estimation | p. 258 |
| Sufficient Statistics | p. 264 |
| Unbiased Estimates Based on a Complete Sufficient Statistic | p. 268 |
| Sampling from a Normal Population | p. 274 |
| The Multidimensional Gaussian Distribution | p. 279 |
| Tables | p. 286 |
| A Brief Bibliography | p. 289 |
| Solutions to Problems | p. 290 |
| Index | p. 333 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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