Probability: With Applications and R

Table of Contents:
Preface xi Acknowledgments xiv Introduction xv 1 First Principles 1 1.1 Random Experiment, Sample Space, Event 1 1.2 What Is a Probability? 3 1.3 Probability Function 4 1.4 Properties of Probabilities 7 1.5 Equally Likely Outcomes 10 1.6 Counting I 12 1.7 Problem-Solving Strategies: Complements, Inclusion–Exclusion 14 1.8 Random Variables 18 1.9 A Closer Look at Random Variables 21 1.10 A First Look at Simulation 22 1.11 Summary 26 Exercises 27 2 Conditional Probability 34 2.1 Conditional Probability 34 2.2 New Information Changes the Sample Space 39 2.3 Finding P(A and B) 40 2.4 Conditioning and the Law of Total Probability 49 2.5 Bayes Formula and Inverting a Conditional Probability 57 2.6 Summary 61 Exercises 62 3 Independence and Independent Trials 68 3.1 Independence and Dependence 68 3.2 Independent Random Variables 76 3.3 Bernoulli Sequences 77 3.4 Counting II 79 3.5 Binomial Distribution 88 3.6 Stirling’s Approximation 95 3.7 Poisson Distribution 96 3.8 Product Spaces 105 3.9 Summary 107 Exercises 109 4 Random Variables 117 4.1 Expectation 118 4.2 Functions of Random Variables 121 4.3 Joint Distributions 125 4.4 Independent Random Variables 130 4.5 Linearity of Expectation 135 4.6 Variance and Standard Deviation 140 4.7 Covariance and Correlation 149 4.8 Conditional Distribution 156 4.9 Properties of Covariance and Correlation 162 4.10 Expectation of a Function of a Random Variable 164 4.11 Summary 165 Exercises 168 5 A Bounty of Discrete Distributions 176 5.1 Geometric Distribution 176 5.2 Negative Binomial—Up from the Geometric 184 5.3 Hypergeometric—Sampling Without Replacement 189 5.4 From Binomial to Multinomial 194 5.5 Benford’s Law 201 5.6 Summary 203 Exercises 205 6 Continuous Probability 211 6.1 Probability Density Function 213 6.2 Cumulative Distribution Function 216 6.3 Uniform Distribution 220 6.4 Expectation and Variance 222 6.5 Exponential Distribution 224 6.6 Functions of Random Variables I 229 6.7 Joint Distributions 235 6.8 Independence 243 6.9 Covariance, Correlation 249 6.10 Functions of Random Variables II 251 6.11 Geometric Probability 256 6.12 Summary 262 Exercises 265 7 Continuous Distributions 273 7.1 Normal Distribution 273 7.2 Gamma Distribution 290 7.3 Poisson Process 296 7.4 Beta Distribution 304 7.5 Pareto Distribution, Power Laws, and the 80-20 Rule 308 7.6 Summary 312 Exercises 315 8 Conditional Distribution, Expectation, and Variance 322 8.1 Conditional Distributions 322 8.2 Discrete and Continuous: Mixing it up 328 8.3 Conditional Expectation 332 8.4 Computing Probabilities by Conditioning 342 8.5 Conditional Variance 346 8.6 Summary 352 Exercises 353 9 Limits 359 9.1 Weak Law of Large Numbers 361 9.2 Strong Law of Large Numbers 367 9.3 Monte Carlo Integration 372 9.4 Central Limit Theorem 376 9.5 Moment-Generating Functions 385 9.6 Summary 391 Exercises 392 10 Additional Topics 399 10.1 Bivariate Normal Distribution 399 10.2 Transformations of Two Random Variables 407 10.3 Method of Moments 411 10.4 Random Walk on Graphs 413 10.5 Random Walks on Weighted Graphs and Markov Chains 421 10.6 From Markov Chain to Markov Chain Monte Carlo 429 10.7 Summary 440 Exercises 442 Appendix A Getting Started with R 447 Appendix B Probability Distributions in R 458 Appendix C Summary of Probability Distributions 459 Appendix D Reminders from Algebra and Calculus 462 Appendix E More Problems for Practice 464 Solutions to Exercises 469 References 487 Index 491

About the Author :
ROBERT P. DOBROW, PhD, is Professor of Mathematics at Carleton College. He has taught probability for over fifteen years and has authored numerous papers in probability theory, Markov chains, and statistics. ROBERT P. DOBROW, PhD, is Professor of Mathematics at Carleton College. He has taught probability for over fifteen years and has authored numerous papers in probability theory, Markov chains, and statistics.