Probability: With Applications and R

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Cover Title Page Copyright Contents Preface Acknowledgments About the Companion Website Introduction Chapter 1 First Principles 1.1 Random Experiment, Sample Space, Event 1.2 What Is a Probability? 1.3 Probability Function 1.4 Properties of Probabilities 1.5 Equally likely outcomes 1.6 Counting I 1.6.1 Permutations 1.7 Counting II 1.7.1 Combinations and Binomial Coefficients 1.8 Problem‐Solving Strategies: Complements and Inclusion–Exclusion 1.9 A First Look at Simulation 1.10 Summary Exercises Chapter 2 Conditional Probability and Independence 2.1 Conditional Probability 2.2 New Information Changes the Sample Space 2.3 Finding P(A and B) 2.3.1 Birthday Problem 2.4 Conditioning and the Law of Total Probability 2.5 Bayes Formula and Inverting a Conditional Probability 2.6 Independence and Dependence 2.7 Product Spaces* 2.8 Summary Exercises Chapter 3 INTRODUCTION TO DISCRETE RANDOM VARIABLES Learning Outcomes 3.1 Random Variables 3.2 Independent Random Variables 3.3 Bernoulli Sequences 3.4 Binomial Distribution 3.5 Poisson Distribution 3.5.1 Poisson Approximation of Binomial Distribution 3.5.2 Poisson as Limit of Binomial Probabilities* 3.6 Summary Exercises Chapter 4 Expectation and More with Discrete Random Variables 4.1 Expectation 4.2 Functions of Random Variables 4.3 Joint distributions 4.4 Independent Random Variables 4.4.1 Sums of Independent Random Variables 4.5 Linearity of expectation 4.6 Variance and Standard Deviation 4.7 Covariance and Correlation 4.8 Conditional Distribution 4.8.1 Introduction to Conditional Expectation 4.9 Properties of Covariance and Correlation* 4.10 Expectation of a Function of a Random Variable* 4.11 Summary Exercises Chapter 5 More Discrete Distributions and Their Relationships 5.1 Geometric Distribution 5.1.1 Memorylessness 5.1.2 Coupon Collecting and Tiger Counting 5.2 Moment‐Generating Functions 5.3 Negative Binomial—Up from the Geometric 5.4 Hypergeometric—Sampling Without Replacement 5.5 From Binomial to Multinomial 5.6 Benford's Law* 5.7 Summary Exercises Chapter 6 Continuous Probability 6.1 Probability Density Function 6.2 Cumulative Distribution Function 6.3 Expectation and Variance 6.4 Uniform Distribution 6.5 Exponential Distribution 6.5.1 Memorylessness 6.6 Joint Distributions 6.7 Independence 6.7.1 Accept–Reject Method 6.8 Covariance, Correlation 6.9 Summary Exercises Chapter 7 Continuous Distributions 7.1 Normal Distribution 7.1.1 Standard Normal Distribution 7.1.2 Normal Approximation of Binomial Distribution 7.1.3 Quantiles 7.1.4 Sums of Independent Normals 7.2 Gamma Distribution 7.2.1 Probability as a Technique of Integration 7.3 Poisson Process 7.4 Beta Distribution 7.5 Pareto Distribution* 7.6 Summary Exercises Chapter 8 Densities of Functions of Random Variables 8.1 Densities via CDFs 8.1.1 Simulating a Continuous Random Variable 8.1.2 Method of Transformations 8.2 Maximums, Minimums, and Order Statistics 8.3 Convolution 8.4 Geometric Probability 8.5 Transformations of Two Random Variables* 8.6 Summary Exercises Chapter 9 Conditional Distribution, Expectation, and Variance 9.1 Conditional Distributions 9.2 DISCRETE AND CONTINUOUS: MIXING IT UP 9.3 CONDITIONAL EXPECTATION 9.3.1 From Function to Random Variable 9.3.2 Random Sum of Random Variables 9.4 COMPUTING PROBABILITIES BY CONDITIONING 9.5 CONDITIONAL VARIANCE 9.6 BIVARIATE NORMAL DISTRIBUTION* 9.7 SUMMARY Exercises Chapter 10 Limits 10.1 WEAK LAW OF LARGE NUMBERS 10.1.1 Markov and Chebyshev Inequalities 10.2 STRONG LAW OF LARGE NUMBERS 10.3 METHOD OF MOMENTS* 10.4 MONTE CARLO INTEGRATION 10.5 CENTRAL LIMIT THEOREM 10.5.1 Central Limit Theorem and Monte Carlo 10.6 A PROOF OF THE CENTRAL LIMIT THEOREM 10.7 SUMMARY Exercises Chapter 11 Beyond Random Walks And Markov Chains 11.1 RANDOM WALKS ON GRAPHS 11.1.1 Long‐Term Behavior 11.2 RANDOM WALKS ON WEIGHTED GRAPHS AND MARKOV CHAINS 11.2.1 Stationary Distribution 11.3 FROM MARKOV CHAIN TO MARKOV CHAIN MONTE CARLO 11.4 SUMMARY Exercises Chapter A Probability Distributions in R Chapter B Summary of Probability Distributions Chapter C Mathematical Reminders Chapter D Working with Joint Distributions SOLUTIONS TO EXERCISES References Index EULA