Introduction to probability with R

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.

Foreword	p. xi
Preface	p. xiii
1 Sets, Events and Probability	p. 1
1.1 The Algebra of Sets	p. 2
1.2 The Bernoulli Sample Space	p. 5
1.3 The Algebra of Multisets	p. 7
1.4 The Concept of Probability	p. 8
1.5 Properties of Probability Measures	p. 9
1.6 Independent Events	p. 11
1.7 The Bernoulli Process	p. 12
1.8 The R Language	p. 14
1.9 Exercises	p. 19
1.10 Answers to Selected Exercises	p. 22
2 Finite Processes	p. 29
2.1 The Basic Models	p. 30
2.2 Counting Rules	p. 31
2.3 Computing Factorials	p. 32
2.4 The Second Rule of Counting	p. 33
2.5 Computing Probabilities	p. 35
2.6 Exercises	p. 38
2.7 Answers to Selected Exercises	p. 42
3 Discrete Random Variables	p. 47
3.1 The Bernoulli Process: Tossing a Coin	p. 49
3.2 The Bernoulli Process: Random Walk	p. 61
3.3 Independence and Joint Distributions	p. 62
3.4 Expectations	p. 64
3.5 The Inclusion-Exclusion Principle	p. 67
3.6 Exercises	p. 71
3.7 Answers to Selected Exercises	p. 75
4 General Random Variables	p. 87
4.1 Order Statistics	p. 91
4.2 The Concept of a General Random Variable	p. 93
4.3 Joint Distribution and Joint Density	p. 96
4.4 Mean, Median and Mode	p. 97
4.5 The Uniform Process	p. 98
4.6 Table of Probability Distributions	p. 102
4.7 Scale Invariance	p. 104
4.8 Exercises	p. 106
4.9 Answers to Selected Exercises	p. 111
5 Statistics and the Normal Distribution	p. 119
5.1 Variance	p. 120
5.2 Bell-Shaped Curve	p. 126
5.3 The Central Limit Theorem	p. 128
5.4 Significance Levels	p. 132
5.5 Confidence Intervals	p. 134
5.6 The Law of Large Numbers	p. 137
5.7 The Cauchy Distribution	p. 139
5.8 Exercises	p. 143
5.9 Answers to Selected Exercises	p. 153
6 Conditional Probability	p. 165
6.1 Discrete Conditional Probability	p. 166
6.2 Gaps and Runs in the Bernoulli Process	p. 170
6.3 Sequential Sampling	p. 173
6.4 Continuous Conditional Probability	p. 177
6.5 Conditional Densities	p. 180
6.6 Gaps in the Uniform Process	p. 182
6.7 The Algebra of Probability Distributions	p. 186
6.8 Exercises	p. 191
6.9 Answers to Selected Exercises	p. 199
7 The Poisson Process	p. 209
7.1 Continuous Waiting Times	p. 209
7.2 Comparing Bernoulli with Uniform	p. 215
7.3 The Poisson Sample Space	p. 220
7.4 Consistency of the Poisson Process	p. 228
7.5 Exercises	p. 229
7.6 Answers to Selected Exercises	p. 235
8 Randomization and Compound Processes	p. 241
8.1 Randomized Bernoulli Process	p. 242
8.2 Randomized Uniform Process	p. 243
8.3 Randomized Poisson Process	p. 245
8.4 Laplace Transforms and Renewal Processes	p. 247
8.5 Proof of the Central Limit Theorem	p. 251
8.6 Randomized Sampling Processes	p. 252
8.7 Prior and Posterior Distributions	p. 253
8.8 Reliability Theory	p. 256
8.9 Bayesian Networks	p. 259
8.10 Exercises	p. 263
8.11 Answers to Selected Exercises	p. 266
9 Entropy and Information	p. 275
9.1 Discrete Entropy	p. 275
9.2 The Shannon Coding Theorem	p. 282
9.3 Continuous Entropy	p. 285
9.4 Proofs of Shannon's Theorems	p. 292
9.5 Exercises	p. 297
9.6 Answers to Selected Exercises	p. 298
10 Markov Chains	p. 303
10.1 The Markov Property	p. 303
10.2 The Ruin Problem	p. 307
10.3 The Network of a Markov Chain	p. 312
10.4 The Evolution of a Markov Chain	p. 314
10.5 The Markov Sample Space	p. 318
10.6 Invariant Distributions	p. 322
10.7 Monte Carlo Markov Chains	p. 327
10.8 Exercises	p. 330
10.9 Answers to Selected Exercises	p. 332
A Random Walks	p. 343
A.1 Fluctuations of Random Walks	p. 343
A.2 The Arcsine Law of Random Walks	p. 347
B Memorylessness and Scale-Invariance	p. 351
B.1 Memorylessness	p. 351
B.2 Self-Similarity	p. 352
References	p. 355
Index	p. 357

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