Probability, Statistics, and Stochastic Processes

PETER OLOFSSON, PhD, is Professor in the Mathematics Department at Trinity University. Dr. Olofsson's research interests include stochastic processes, branching processes, mathematical biology, and Poisson approximation. He is the author of Probabilities: The Little Numbers That Rule Our Lives, also published by Wiley.MIKAEL ANDERSSON, PhD, is Associate Professor in the Department of Applied Statistics at the Swedish University of Agricultural Sciences. Dr. Andersson's research interests include stochastic modeling of infectious diseases, epidemiology, and biological applications.

• Preface xiPreface to the First Edition xiii1 Basic Probability Theory 11.1 Introduction 11.2 Sample Spaces and Events 31.3 The Axioms of Probability 71.4 Finite Sample Spaces and Combinatorics 151.4.1 Combinatorics 171.5 Conditional Probability and Independence 271.6 The Law of Total Probability and Bayes’ Formula 41Problems 632 Random Variables 762.1 Introduction 762.2 Discrete Random Variables 772.3 Continuous Random Variables 822.4 Expected Value and Variance 952.5 Special Discrete Distributions 1112.6 The Exponential Distribution 1232.7 The Normal Distribution 1272.8 Other Distributions 1312.9 Location Parameters 1372.10 The Failure Rate Function 139Problems 1443 Joint Distributions 1563.1 Introduction 1563.2 The Joint Distribution Function 1563.3 Discrete Random Vectors 1583.4 Jointly Continuous Random Vectors 1603.5 Conditional Distributions and Independence 1643.5.1 Independent Random Variables 1683.6 Functions of Random Vectors 1723.7 Conditional Expectation 1853.8 Covariance and Correlation 1963.9 The Bivariate Normal Distribution 2093.10 Multidimensional Random Vectors 2163.11 Generating Functions 2313.12 The Poisson Process 240Problems 2474 Limit Theorems 2634.1 Introduction 2634.2 The Law of Large Numbers 2644.3 The Central Limit Theorem 2684.4 Convergence in Distribution 275Problems 2785 Simulation 2815.1 Introduction 2815.2 Random Number Generation 2825.3 Simulation of Discrete Distributions 2835.4 Simulation of Continuous Distributions 2855.5 Miscellaneous 290Problems 2926 Statistical Inference 2946.1 Introduction 2946.2 Point Estimators 2946.3 Confidence Intervals 3046.4 Estimation Methods 3126.5 Hypothesis Testing 3276.6 Further Topics in Hypothesis Testing 3346.7 Goodness of Fit 3396.8 Bayesian Statistics 3516.9 Nonparametric Methods 363Problems 3787 Linear Models 3917.1 Introduction 3917.2 Sampling Distributions 3927.3 Single Sample Inference 3957.4 Comparing Two Samples 4027.5 Analysis of Variance 4097.6 Linear Regression 4157.7 The General Linear Model 431Problems 4368 Stochastic Processes 4448.1 Introduction 4448.2 Discrete -Time Markov Chains 4458.3 Random Walks and Branching Processes 4648.4 Continuous -Time Markov Chains 4758.5 Martingales 4948.6 Renewal Processes 5028.7 Brownian Motion 509Problems 517Appendix A Tables 527Appendix B Answers to Selected Problems 535Further Reading 551Index 553