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Understanding Probability, Third Edition

Book Description

Understanding Probability is a unique and stimulating approach to a first course in probability. The first part of the book demystifies probability and uses many wonderful probability applications from everyday life to help the reader develop a feel for probabilities. The second part, covering a wide range of topics, teaches clearly and simply the basics of probability. This fully revised third edition has been packed with even more exercises and examples and it includes new sections on Bayesian inference, Markov chain Monte-Carlo simulation, hitting probabilities in random walks and Brownian motion, and a new chapter on continuous-time Markov chains with applications. Here you will find all the material taught in an introductory probability course. The first part of the book, with its easy-going style, can be read by anybody with a reasonable background in high school mathematics. The second part of the book requires a basic course in calculus.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Introduction
  7. Part One: Probability in Action
    1. Chapter 1: Probability Questions
    2. Chapter 2: Law of Large Numbers and Simulation
      1. 2.1 Law of Large Numbers for Probabilities
      2. 2.2 Basic Probability Concepts
      3. 2.3 Expected Value and The Law of Large Numbers
      4. 2.4 Drunkard’s Walk
      5. 2.5 St. Petersburg Paradox
      6. 2.6 Roulette and The Law of Large Numbers
      7. 2.7 Kelly Betting System
      8. 2.8 Random-number Generator
      9. 2.9 Simulating from Probability Distributions
      10. 2.10 Problems
    3. Chapter 3: Probabilities in Everyday Life
      1. 3.1 Birthday Problem
      2. 3.2 Coupon Collector’s Problem
      3. 3.3 Craps
      4. 3.4 Gambling Systems for Roulette
      5. 3.5 Gambler’s Ruin Problem
      6. 3.6 Optimal Stopping
      7. 3.7 The 1970 Draft Lottery
      8. 3.8 Problems
    4. Chapter 4: Rare Events and Lotteries
      1. 4.1 Binomial Distribution
      2. 4.2 Poisson Distribution
      3. 4.3 Hypergeometric Distribution
      4. 4.4 Problems
    5. Chapter 5: Probability and Statistics
      1. 5.1 Normal Curve
      2. 5.2 Concept of Standard Deviation
      3. 5.3 Square-Root Law
      4. 5.4 Central Limit Theorem
      5. 5.5 Graphical Illustration of the Central Limit Theorem
      6. 5.6 Statistical Applications
      7. 5.7 Confidence Intervals for Simulations
      8. 5.8 Central Limit Theorem and Random Walks
      9. 5.9 Brownian Motion
      10. 5.10 Falsified Data and Benford’s Law
      11. 5.11 Normal Distribution Strikes Again
      12. 5.12 Statistics and Probability Theory
      13. 5.13 Problems
    6. Chapter 6: Chance Trees and Bayes’ Rule
      1. 6.1 Monty Hall Dilemma
      2. 6.2 Test Paradox
      3. 6.3 Problems
  8. Part Two: Essentials of Probability
    1. Chapter 7: Foundations of Probability Theory
      1. 7.1 Probabilistic Foundations
      2. 7.2 Compound Chance Experiments
      3. 7.3 Some Basic Rules
    2. Chapter 8: Conditional Probability and Bayes
      1. 8.1 Conditional Probability
      2. 8.2 Law of Conditional Probability
      3. 8.3 Bayes’ Rule in Odds Form
      4. 8.4 Bayesian Statistics – Discrete Case
    3. Chapter 9: Basic Rules for Discrete Random Variables
      1. 9.1 Random Variables
      2. 9.2 Expected Value
      3. 9.3 Expected Value of Sums of Random Variables
      4. 9.4 Substitution Rule and Variance
      5. 9.5 Independence of Random Variables
      6. 9.6 Important Discrete Random Variables
    4. Chapter 10: Continuous Random Variables
      1. 10.1 Concept of Probability Density
      2. 10.2 Expected Value of A Continuous Random Variable
      3. 10.3 Substitution Rule and The Variance
      4. 10.4 Important Probability Densities
      5. 10.5 Transformation of Random Variables
      6. 10.6 Failure Rate Function
    5. Chapter 11: Jointly Distributed Random Variables
      1. 11.1 Joint Probability Mass Function
      2. 11.2 Joint Probability Density Function
      3. 11.3 Marginal Probability Densities
      4. 11.4 Transformation of Random Variables
      5. 11.5 Covariance and Correlation Coefficient
    6. Chapter 12: Multivariate Normal Distribution
      1. 12.1 Bivariate Normal Distribution
      2. 12.2 Multivariate Normal Distribution
      3. 12.3 Multidimensional Central Limit Theorem
      4. 12.4 Chi-Square Test
    7. Chapter 13: Conditioning by Random Variables
      1. 13.1 Conditional Distributions
      2. 13.2 Law of Conditional Probability for Random Variables
      3. 13.3 Law of Conditional Expectation
      4. 13.4 Conditional Expectation as a Computational Tool
      5. 13.5 Bayesian Statistics – Continuous Case
    8. Chapter 14: Generating Functions
      1. 14.1 Generating Functions
      2. 14.2 Moment-Generating Functions
      3. 14.3 Chernoff Bound
      4. 14.4 Strong Law of Large Numbers Revisited
      5. 14.5 Central Limit Theorem Revisited
      6. 14.6 Law of The Iterated Logarithm
    9. Chapter 15: Discrete-Time Markov Chains
      1. 15.1 Markov Chain Model
      2. 15.2 Time-Dependent Analysis of Markov Chains
      3. 15.3 Absorbing Markov Chains
      4. 15.4 Long-Run Analysis of Markov Chains
      5. 15.5 Markov Chain Monte Carlo Simulation
    10. Chapter 16: Continuous-Time Markov Chains
      1. 16.1 Markov Chain Model
      2. 16.2 Time-Dependent Probabilities
      3. 16.3 Limiting Probabilities
  9. Appendix Counting Methods and e[sup(x)]
  10. Recommended Reading
  11. Answers to Odd-Numbered Problems
  12. Bibliography
  13. Index