Introduction to Probability with R

ISBN-13: 9781420065213 / Angielski / Twarda / 2008 / 380 str.

Kenneth P. Baclawski; Jim Zidek; Bradley P. Carlin

Opis

Based on a popular course taught by the late Gian-Carlo Rota of MIT, with many new topics covered as well, Introduction to Probability with R presents R programs and animations to provide an intuitive yet rigorous understanding of how to model natural phenomena from a probabilistic point of view. Although the R programs are small in length, they are just as sophisticated and powerful as longer programs in other languages. This brevity makes it easy for students to become proficient in R. This calculus-based introduction organizes the material around key themes. One of the most important themes centers on viewing probability as a way to look at the world, helping students think and reason probabilistically. The text also shows how to combine and link stochastic processes to form more complex processes that are better models of natural phenomena. In addition, it presents a unified treatment of transforms, such as Laplace, Fourier, and z; the foundations of fundamental stochastic processes using entropy and information; and an introduction to Markov chains from various viewpoints. Each chapter includes a short biographical note about a contributor to probability theory, exercises, and selected answers. The book has an accompanying website with more information.

Spis treści

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.

O autorze

Kenneth Baclawski