PROBABILITY. Wiley. With Applications and R ROBERT P. DOBROW. Department of Mathematics. Carleton College Northfield, MN

Transcription

1 PROBABILITY With Applications and R ROBERT P. DOBROW Department of Mathematics Carleton College Northfield, MN Wiley

2 CONTENTS Preface Acknowledgments Introduction xi xiv xv 1 First Principles Random Experiment, Sample Space, Event What Is a Probability? Probability Function Properties of Probabilities Equally Likely Outcomes Counting I Problem-Solving Strategies: Complements, Inclusion-Exclusion Random Variables A Closer Look at Random Variables A First Look at Simulation Summary 26 Exercises 27 2 Conditional Probability Conditional Probability New Information Changes the Sample Space Finding P(A and B) Birthday Problem 45 vii

3 viii CONTENTS 2.4 Conditioning and the Law of Total Probability Bayes Formula and Inverting a Conditional Probability Summary 61 Exercises 62 3 Independence and Independent Trials Independence and Dependence Independent Random Variables Bernoulli Sequences Counting II Binomial Distribution Stirling's Approximation Poisson Distribution Poisson Approximation of Binomial Distribution Poisson Limit Product Spaces Summary 107 Exercises Random Variables Expectation Functions of Random Variables Joint Distributions Independent Random Variables Sums of Independent Random Variables Linearity of Expectation Indicator Random Variables Variance and Standard Deviation Covariance and Correlation Conditional Distribution Introduction to Conditional Expectation Properties of Covariance and Correlation Expectation of a Function of a Random Variable Summary 165 Exercises A Bounty of Discrete Distributions Geometric Distribution Memory lessness Coupon Collecting and Tiger Counting How R Codes the Geometric Distribution Negative Binomial Up from the Geometric Hypergeometric Sampling Without Replacement 189

4 CONTENTS ix 5.4 From Binomial to Multinomial Multinomial Counts Benford's Law Summary 203 Exercises Continuous Probability Probability Density Function Cumulative Distribution Function Uniform Distribution Expectation and Variance Exponential Distribution Memorylessness Functions of Random Variables I Simulating a Continuous Random Variable Joint Distributions Independence Accept-Reject Method Covariance, Correlation Functions of Random Variables II Maximums and Minimums Sums of Random Variables Geometric Probability Summary 262 Exercises Continuous Distributions Normal Distribution Standard Normal Distribution Normal Approximation of Binomial Distribution Sums of Independent Normals Gamma Distribution Probability as a Technique of Integration Sum of Independent Exponentials Poisson Process Beta Distribution Pareto Distribution, Power Laws, and the Rule Simulating the Pareto Distribution Summary 312 Exercises Conditional Distribution, Expectation, and Variance Conditional Distributions Discrete and Continuous: Mixing it up 328

5 X CONTENTS 8.3 Conditional Expectation From Function to Random Variable Random Sum of Random Variables Computing Probabilities by Conditioning Conditional Variance Summary 352 Exercises Limits Weak Law of Large Numbers Markov and Chebyshev Inequalities Strong Law of Large Numbers Monte Carlo Integration Central Limit Theorem Central Limit Theorem and Monte Carlo Moment-Generating Functions Proof of Central Limit Theorem Summary 391 Exercises Additional Topics Bivariate Normal Distribution Transformations of Two Random Variables Method of Moments Random Walk on Graphs Long-Term Behavior Random Walks on Weighted Graphs and Markov Chains Stationary Distribution From Markov Chain to Markov Chair* Monte Carlo 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