COPYRIGHTED MATERIAL CONTENTS. Preface Preface to the First Edition

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1 Preface Preface to the First Edition xi xiii 1 Basic Probability Theory Introduction Sample Spaces and Events The Axioms of Probability Finite Sample Spaces and Combinatorics Combinatorics Conditional Probability and Independence Independent Events The Law of Total Probability and Bayes Formula Bayes Formula Genetics and Probability Recursive Methods 55 Problems 63 COPYRIGHTED MATERIAL 2 Random Variables Introduction Discrete Random Variables Continuous Random Variables The Uniform Distribution Functions of Random Variables Expected Value and Variance The Expected Value of a Function of a Random Variable Variance of a Random Variable 104 v

2 vi 2.5 Special Discrete Distributions Indicators The Binomial Distribution The Geometric Distribution The Poisson Distribution The Hypergeometric Distribution Describing Data Sets The Exponential Distribution The Normal Distribution Other Distributions The Lognormal Distribution The Gamma Distribution The Cauchy Distribution Mixed Distributions Location Parameters The Failure Rate Function Uniqueness of the Failure Rate Function 141 Problems Joint Distributions Introduction The Joint Distribution Function Discrete Random Vectors Jointly Continuous Random Vectors Conditional Distributions and Independence Independent Random Variables Functions of Random Vectors Real-Valued Functions of Random Vectors The Expected Value and Variance of a Sum Vector-Valued Functions of Random Vectors Conditional Expectation Conditional Expectation as a Random Variable Conditional Expectation and Prediction Conditional Variance Recursive Methods Covariance and Correlation The Correlation Coefficient The Bivariate Normal Distribution Multidimensional Random Vectors Order Statistics Reliability Theory The Multinomial Distribution The Multivariate Normal Distribution Convolution 227

3 vii 3.11 Generating Functions The Probability Generating Function The Moment Generating Function The Poisson Process Thinning and Superposition 244 Problems Limit Theorems Introduction The Law of Large Numbers The Central Limit Theorem The Delta Method Convergence in Distribution Discrete Limits Continuous Limits 277 Problems Simulation Introduction Random Number Generation Simulation of Discrete Distributions Simulation of Continuous Distributions Miscellaneous 290 Problems Statistical Inference Introduction Point Estimators Estimating the Variance Confidence Intervals Confidence Interval for the Mean in the Normal Distribution with Known Variance Confidence Interval for an Unknown Probability One-Sided Confidence Intervals Estimation Methods The Method of Moments Maximum Likelihood Evaluation of Estimators with Simulation Bootstrap Simulation Hypothesis Testing Large Sample Tests Test for an Unknown Probability 333

4 viii 6.6 Further Topics in Hypothesis Testing P-Values Data Snooping The Power of a Test Multiple Hypothesis Testing Goodness of Fit Goodness-of-Fit Test for Independence Fisher s Exact Test Bayesian Statistics Noninformative priors Credibility Intervals Nonparametric Methods Nonparametric Hypothesis Testing Comparing Two Samples Nonparametric Confidence Intervals 375 Problems Linear Models Introduction Sampling Distributions Single Sample Inference Inference for the Variance Inference for the Mean Comparing Two Samples Inference about Means Inference about Variances Analysis of Variance One-Way Analysis of Variance Multiple Comparisons: Tukey s Method Kruskal Wallis Test Linear Regression Prediction Goodness of Fit The Sample Correlation Coefficient Spearman s Correlation Coefficient The General Linear Model 431 Problems Stochastic Processes Introduction Discrete -Time Markov Chains Time Dynamics of a Markov Chain Classification of States 450

5 ix Stationary Distributions Convergence to the Stationary Distribution Random Walks and Branching Processes The Simple Random Walk Multidimensional Random Walks Branching Processes Continuous -Time Markov Chains Stationary Distributions and Limit Distributions Birth Death Processes Queueing Theory Further Properties of Queueing Systems Martingales Martingale Convergence Stopping Times Renewal Processes Asymptotic Properties Brownian Motion Hitting Times Variations of the Brownian Motion 515 Problems 517 Appendix A Tables 527 Appendix B Answers to Selected Problems 535 Further Reading 551 Index 553

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