Introduction to Probability With R

Based on the popular probability course by Gian-Carlo Rota of MIT, Probability and Random Processes with R provides a calculus-based introduction to probability. The text systemically motivates and organizes the standard distributions that most often occur in probability using physical processes. Presenting a probabilistic approach that builds on other approaches such as geometry and physical processes, the book addresses sets, events, and probability; finite processes; random variables; statistics and normal distribution; conditional probability; the Poisson process; entropy and information; Markov chains; Markov processes; Bayesian networks; and the Bayesian web. Various exercises and examples compare different perspectives.

FOREWORD PREFACE Sets, Events, and Probability The Algebra of Sets The Bernoulli Sample Space The Algebra of Multisets The Concept of Probability Properties of Probability Measures Independent Events The Bernoulli Process The R Language Finite Processes The Basic Models Counting Rules Computing Factorials The Second Rule of Counting Computing Probabilities Discrete Random Variables The Bernoulli Process: Tossing a Coin The Bernoulli Process: Random Walk Independence and Joint Distributions Expectations The Inclusion-Exclusion Principle General Random Variables Order Statistics The Concept of a General Random Variable Joint Distribution and Joint Density Mean, Median and Mode The Uniform Process Table of Probability Distributions Scale Invariance Statistics and the Normal Distribution Variance Bell-Shaped Curve The Central Limit Theorem Significance Levels Confidence Intervals The Law of Large Numbers The Cauchy Distribution Conditional Probability Discrete Conditional Probability Gaps and Runs in the Bernoulli Process Sequential Sampling Continuous Conditional Probability Conditional Densities Gaps in the Uniform Process The Algebra of Probability Distributions The Poisson Process Continuous Waiting Times Comparing Bernoulli with Uniform The Poisson Sample Space Consistency of the Poisson Process Randomization and Compound Processes Randomized Bernoulli Process Randomized Uniform Process Randomized Poisson Process Laplace Transforms and Renewal Processes Proof of the Central Limit Theorem Randomized Sampling Processes Prior and Posterior Distributions Reliability Theory Bayesian Networks Entropy and Information Discrete Entropy The Shannon Coding Theorem Continuous Entropy Proofs of Shannon's Theorems Markov Chains The Markov Property The Ruin Problem The Network of a Markov Chain The Evolution of a Markov Chain The Markov Sample Space Invariant Distributions Monte Carlo Markov Chains appendix A: Random Walks Fluctuations of Random Walks The Arcsine Law of Random Walks Appendix B: Memorylessness and Scale-Invariance Memorylessness Self-Similarity References Index Exercises and Answers appear at the end of each chapter.