Probability: With Applications and R

ISBN-13: 9781119692386 / Angielski / Twarda / 2021 / 544 str.

Amy S. Wagaman; Robert P. Dobrow

Spis treści

Preface xiiiAcknowledgments xviiIntroduction xix1 First Principles 11.1 Random Experiment, Sample Space, Event 11.2 What Is a Probability? 31.3 Probability Function 41.4 Properties of Probabilities 71.5 Equally Likely Outcomes 111.6 Counting I 121.6.1 Permutations 131.7 Counting II 161.7.1 Combinations and Binomial Coefficients 171.8 Problem-Solving Strategies: Complements andInclusion-Exclusion 261.9 A First Look at Simulation 291.10 Summary 34Exercises 362 Conditional Probability and Independence 452.1 Conditional Probability 452.2 New Information Changes the Sample Space 502.3 Finding P (A and B) 512.3.1 Birthday Problem 562.4 Conditioning and the Law of Total Probability 602.5 Bayes Formula and Inverting a Conditional Probability 672.6 Independence and Dependence 722.7 Product Spaces 802.8 Summary 82Exercises 833 Introduction to Discrete Random Variables 933.1 Random Variables 933.2 Independent Random Variables 973.3 Bernoulli Sequences 993.4 Binomial Distribution 1013.5 Poisson Distribution 1083.5.1 Poisson Approximation of Binomial Distribution 1133.5.2 Poisson as Limit of Binomial Probabilitie; 1153.6 Summary 116Exercises 1184 Expectation and More with Discrete Random Variables 1254.1 Expectation 1274.2 Functions of Random Variables 1304.3 Joint Distributions 1344.4 Independent Random Variables 1394.4.1 Sums of Independent Random Variables 1424.5 Linearity of Expectation 1444.6 Variance and Standard Deviation 1494.7 Covariance and Correlation 1584.8 Conditional Distribution 1654.8.1 Introduction to Conditional Expectation 1684.9 Properties of Covariance and Correlation 1714.10 Expectation of a Function of a Random Variable 1734.11 Summary 174Exercises 1765 More Discrete Distributions and Their Relationships 1855.1 Geometric Distribution 1855.1.1 Memorylessness 1885.1.2 Coupon Collecting and Tiger Counting 1895.2 Moment-Generating Functions 1935.3 Negative Binomial--Up from the Geometric 1965.4 Hypergeometric--Sampling Without Replacement 2025.5 From Binomial to Multinomial 2075.6 Benford's Law 2135.7 Summary 216Exercises 2186 Continuous Probability 2276.1 Probability Density Function 2296.2 Cumulative Distribution Function 2336.3 Expectation and Variance 2376.4 Uniform Distribution 2396.5 Exponential Distribution 2426.5.1 Memorylessness 2436.6 Joint Distributions 2476.7 Independence 2566.7.1 Accept-Reject Method 2586.8 Covariance, Correlation 2626.9 Summary 264Exercises 2667 Continuous Distributions 2737.1 Normal Distribution 2737.1.1 Standard Normal Distribution 2767.1.2 Normal Approximation of Binomial Distribution 2787.1.3 Quantiles 2827.1.4 Sums of Independent Normals 2857.2 Gamma Distribution 2887.2.1 Probability as a Technique of Integration 2927.3 Poisson Process 2947.4 Beta Distribution 3027.5 Pareto Distribution 3057.6 Summary 308Exercises 3118 Densities of Functions of Random Variables 3198.1 Densities via CDFs 3208.1.1 Simulating a Continuous Random Variable 3268.1.2 Method of Transformations 3298.2 Maximums, Minimums, and Order Statistics 3308.3 Convolution 3358.4 Geometric Probability 3388.5 Transformations of Two Random Variables 3448.6 Summary 348Exercises 3499 Conditional Distribution, Expectation, and Variance 357Introduction 3579.1 Conditional Distributions 3589.2 Discrete and Continuous: Mixing it Up 3649.3 Conditional Expectation 3699.3.1 From Function to Random Variable 3719.3.2 Random Sum of Random Variables 3789.4 Computing Probabilities by Conditioning 3789.5 Conditional Variance 3829.6 Bivariate Normal Distribution 3879.7 Summary 396Exercises 39810 LIMITS 40710.1 Weak Law of Large Numbers 40910.1.1 Markov and Chebyshev Inequalities 41110.2 Strong Law of Large Numbers 41510.3 Method of Moments 42110.4 Monte Carlo Integration 42410.5 Central Limit Theorem 42810.5.1 Central Limit Theorem and Monte Carlo 43610.6 A Proof of the Central Limit Theorem 43710.7 Summary 439Exercises 44011 Beyond Random Walks and Markov Chains 44711.1 Random Walk on Graphs 44711.1.1 Long-Term Behavior 45111.2 Random Walks on Weighted Graphs and Markov Chains 45511.2.1 Stationary Distribution 45811.3 From Markov Chain to Markov Chain Monte Carlo 46211.4 Summary 474Exercises 476Appendix A Probability Distributions in R 481Appendix B Summary of Probability Distributions 483Appendix C Mathematical Reminders 487Appendix D Working with Joint Distributions 489Solutions 497References 511Index 515

O autorze

Amy S. Wagaman, PhD, is Associate Professor of Statistics at Amherst College. She received her doctorate in Statistics at the University of Michigan in 2008. Her research interests include nonparametric statistics, statistics education, dimension reduction and estimation, and covariance estimation and regularization.Robert P. Dobrow, PhD, is Emeritus Professor of Mathematics at Carleton College. He has over 15 years of experience teaching probability and has authored numerous papers in probability theory, Markov chains, and statistics.