Introduction to Probability, Second Edition

Description

Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces. The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment. The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface 1: Probability and counting 1.1 Why study probability? 1.2 Sample spaces and Pebble World 1.3 Naive definition of probability 1.4 How to count 1.5 Story proofs 1.6 Non-naive definition of probability 1.7 Recap 1.8 R 1.9 Exercises 2: Conditional probability 2.1 The importance of thinking conditionally 2.2 Definition and intuition 2.3 Bayes' rule and the law of total probability 2.4 Conditional probabilities are probabilities 2.5 Independence of events 2.6 Coherency of Bayes' rule 2.7 Conditioning as a problem-solving tool 2.8 Pitfalls and paradoxes 2.9 Recap 2.10 R 2.11 Exercises 3: Random variables and their distributions 3.1 Random variables 3.2 Distributions and probability mass functions 3.3 Bernoulli and Binomial 3.4 Hypergeometric 3.5 Discrete Uniform 3.6 Cumulative distribution functions 3.7 Functions of random variables 3.8 Independence of r.v.s 3.9 Connections between Binomial and Hypergeometric 3.10 Recap 3.11 R 3.12 Exercises 4: Expectation 4.1 Definition of expectation 4.2 Linearity of expectation 4.3 Geometric and Negative Binomial 4.4 Indicator r.v.s and the fundamental bridge 4.5 Law of the unconscious statistician (LOTUS) 4.6 Variance 4.7 Poisson 4.8 Connections between Poisson and Binomial 4.9 *Using probability and expectation to prove existence 4.10 Recap 4.11 R 4.12 Exercises 5: Continuous random variables 5.1 Probability density functions 5.2 Uniform 5.3 Universality of the Uniform 5.4 Normal 5.5 Exponential 5.6 Poisson processes 5.7 Symmetry of i.i.d. continuous r.v.s 5.8 Recap 5.9 R 5.10 Exercises 6: Moments 6.1 Summaries of a distribution 6.2 Interpreting moments 6.3 Sample moments 6.4 Moment generating functions 6.5 Generating moments with MGFs 6.6 Sums of independent r.v.s via MGFs 6.7 *Probability generating functions 6.8 Recap 6.9 R 6.10 Exercises 7: Joint distributions 7.1 Joint, marginal, and conditional 7.2 2D LOTUS 7.3 Covariance and correlation 7.4 Multinomial 7.5 Multivariate Normal 7.6 Recap 7.7 R 7.8 Exercises 8: Transformations 8.1 Change of variables 8.2 Convolutions 8.3 Beta 8.4 Gamma 8.5 Beta-Gamma connections 8.6 Order statistics 8.7 Recap 8.8 R 8.9 Exercises 9: Conditional expectation 9.1 Conditional expectation given an event 9.2 Conditional expectation given an r.v. 9.3 Properties of conditional expectation 9.4 *Geometric interpretation of conditional expectation 9.5 Conditional variance 9.6 Adam and Eve examples 9.7 Recap 9.8 R 9.9 Exercises 10: Inequalities and limit theorems 10.1 Inequalities 10.2 Law of large numbers 10.3 Central limit theorem 10.4 Chi-Square and Student-t 10.5 Recap 10.6 R 10.7 Exercises 11: Markov chains 11.1 Markov property and transition matrix 11.2 Classification of states 11.3 Stationary distribution 11.4 Reversibility 11.5 Recap 11.6 R 11.7 Exercises 12: Markov chain Monte Carlo 12.1 Metropolis-Hastings 12.2 Gibbs sampling 12.3 Recap 12.4 R 12.5 Exercises 13: Poisson processes 13.1 Poisson processes in one dimension 13.2 Conditioning, superposition, and thinning 13.3 Poisson processes in multiple dimensions 13.4 Recap 13.5 R 13.6 Exercises A: Math A.1 Sets A.2 Functions A.3 Matrices A.4 Difference equations A.5 Differential equations A.6 Partial derivatives A.7 Multiple integrals A.8 Sums A.9 Pattern recognition A.10 Common sense and checking answers B: R B.1 Vectors B.2 Matrices B.3 Math B.4 Sampling and simulation B.5 Plotting B.6 Programming B.7 Summary statistics B.8 Distributions C: Table of distributions References Index