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Preface 1 1 Sets, Events and Probability 5 1.1 The Algebra of Sets 6 1.2 The Bernoulli Sample Space 9 1.3 The Algebra of Multisets 11 1.4 The Concept of Probability 12 1.5 Properties of Probability Measures 13 1.6 Independent Events 15 1.7 The Bernoulli Process 16 1.8 The R Language 17 1.9 Exercises 23 1.10 Answers to Selected Exercises 26 2 Finite Processes 33 2.1 The Basic Models 34 2.2 Counting Rules 35 2.3 Computing Factorials 36 2.4 The Second Rule of Counting 37 2.5 Computing Probabilities. 39 2.6 Answers to Selected Exercises 46 3 Discrete Random Variables 51 3.1 The Bernoulli Process: Tossing a coin 53 3.2 The Bernoulli Process: Random Walk 64 3.3 Independence and Joint Distributions 65 3.4 Expectations 68 3.5 The Inclusion-Exclusion Principle 70 3.6 Exercises 75 3.7 Answers to Selected Exercises 79 4 General Random Variables 91 4.1 Order Statistics 95 4.2 The Concept of a General Random Variable 97 4.3 Joint Distribution and Joint Density 100 4.4 Mean, Median and Mode 100 i ii CONTENTS 4.5 The Uniform Process 102 4.6 Table of Probability Distributions 106 4.7 Scale Invariance 108 4.8 Exercises 110 4.9 Answers to Selected Exercises 115 5 Statistics and the Normal Distribution 123 5.1 Variance 124 5.2 Bell-Shaped Curve 130 5.3 The Central Limit Theorem 132 5.4 Significance Levels 137 5.5 Confidence Intervals 139 5.6 The Law of Large Numbers 141 5.7 The Cauchy Distribution 144 5.8 Exercises 148 5.9 Answers to Selected Exercises 158 6 Conditional Probability 169 6.1 Discrete Conditional Probability 170 6.2 Gaps and Runs in the Bernoulli Process 174 6.3 Sequential Sampling 177 6.4 Continuous Conditional Probability 181 6.5 Conditional Densities 184 6.6 Gaps in the Uniform Process 186 6.7 The Algebra of Probability Distributions 190 6.8 Exercises 195 6.9 Answers to Selected Exercises 203 7 The Poisson Process 213 7.1 Continuous Waiting Times 213 7.2 Comparing Bernoulli with Uniform 219 7.3 The Poisson Sample Space 224 7.4 Consistency of the Poisson Process 231 7.5 Exercises 233 7.6 Answers to Selected Exercises 239 8 Randomization and Compound Processes 245 8.1 Randomized Bernoulli Process 246 8.2 Randomized Uniform Process 247 8.3 Randomized Poisson Process 249 8.4 Proof of the Central Limit Theorem 251 8.5 Laplace Transforms and Renewal Processes 252 8.6 Randomized Sampling Processes 256 8.7 Prior and Posterior Distributions 257 8.8 Reliability Theory 260 CONTENTS iii 8.9 Bayesian Networks 263 8.10 Exercises 268 8.11 Answers to Selected Exercises 270 9 Entropy and Information 279 9.1 Discrete Entropy 279 9.2 The Shannon Coding Theorem 287 9.3 Continuous Entropy 289 9.4 Proofs of Shannon¿s Theorems 297 9.5 Exercises 301 9.6 Answers to Selected Exercises 303 10 Markov Chains 307 10.1 The Markov Property 307 10.2 The Ruin Problem 311 10.3 The Network of a Markov Chain 316 10.4 The Evolution of a Markov Chain 317 10.5 The Markov Sample Space 322 10.6 Invariant Distributions 325 10.7 Monte Carlo Markov Chains 331 10.8 Exercises 334 10.9 Answers to Selected Exercises 336 A Random Walks 347 A.1 Fluctuations of Random Walks 347 A.2 The Arcsine Law of Random Walks 351 B Memorylessness and Scale-Invariance 355 B.1 Memorylessness 355 B.2 Self-Similarity 356 References 359 Index 360

Library of Congress Subject Headings for this publication:

Probabilities.

R (Computer program language) -- Mathematical models.