Probability: a lively introduction

Description

Probability has applications in many areas of modern science, not to mention in our daily life. Its importance as a mathematical discipline cannot be overrated, and it is a fascinating and surprising topic in its own right. This engaging textbook with its easy-to-follow writing style provides a comprehensive yet concise introduction to the subject. It covers all of the standard material for undergraduate and first-year-graduate-level courses as well as many topics that are usually not found in standard texts, such as Bayesian inference, Markov chain Monte Carlo simulation, and Chernoff bounds. Tijms H. Probability: A Lively Introduction Title Contents Preface 1 Foundations of Probability Theory 1.1 Probabilistic Foundations 1.2 Classical Probability Model 1.3 Geometric Probability Model 1.4 Compound Chance Experiments 1.5 Some Basic Rules 1.6 Inclusion–Exclusion Rule 2 Conditional Probability 2.1 Concept of Conditional Probability 2.2 Chain Rule for Conditional Probabilities 2.3 Law of Conditional Probability 2.4 Bayes’ Rule in Odds Form 2.5 Bayesian Inference − Discrete Case 3 Discrete Random Variables 3.1 Concept of a Random Variable 3.2 Expected Value 3.3 Expected Value of Sums of Random Variables 3.4 Substitution Rule and Variance 3.5 Independence of Random Variables 3.6 Binomial Distribution 3.7 Poisson Distribution 3.8 Hypergeometric Distribution 3.9 Other Discrete Distributions 4 Continuous Random Variables 4.1 Concept of Probability Density 4.2 Expected Value of a Continuous Random Variable 4.3 Substitution Rule and the Variance 4.4 Uniform and Triangular Distributions 4.5 Exponential Distribution 4.6 Gamma, Weibull, and Beta Distributions 4.7 Normal Distribution 4.8 Other Continuous Distributions 4.9 Inverse-Transformation Method and Simulation 4.10 Failure-Rate Function 4.11 Probability Distributions and Entropy 5 Jointly Distributed Random Variables 5.1 Joint Probability Mass Function 5.2 Joint Probability Density Function 5.3 Marginal Probability Densities 5.4 Transformation of Random Variables 5.5 Covariance and Correlation Coefficient 6 Multivariate Normal Distribution 6.1 Bivariate Normal Distribution 6.2 Multivariate Normal Distribution 6.3 Multidimensional Central Limit Theorem 6.4 Chi-Square Test 7 Conditioning by Random Variables 7.1 Conditional Distributions 7.2 Law of Conditional Probability for Random Variables 7.3 Law of Conditional Expectation 7.4 Conditional Expectation as a Computational Tool 7.5 Bayesian Inference − Continuous Case 8 Generating Functions 8.1 Generating Functions 8.2 Branching Processes and Generating Functions 8.3 Moment-Generating Functions 8.4 Central Limit Theorem Revisited 9 Additional Topics in Probability 9.1 Bounds and Inequalities 9.2 Strong Law of Large Numbers 9.3 Kelly Betting System 9.4 Renewal–Reward Processes 10 Discrete-Time Markov Chains 10.1 Markov Chain Model 10.2 Time-Dependent Analysis of Markov Chains 10.3 Absorbing Markov Chains 10.4 Long-Run Analysis of Markov Chains 10.5 Markov Chain Monte Carlo Simulation 11 Continuous-Time Markov Chains 11.1 Markov Chain Model 11.2 Time-Dependent Probabilities 11.3 Limiting Probabilities Appendix A: Counting Methods Appendix B: Basics of Set Theory Appendix C: Some Basic Results from Calculus Appendix D: Basics of Monte Carlo Simulation Answers to Odd-Numbered Problems Index