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| Preface | |
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| Combinatorial Analysis | |
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| Introduction | |
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| The Basic Principle of Counting | |
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| Permutations | |
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| Combinations | |
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| Multinomial Coefficients | |
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| The Number of Integer Solutions of Equations | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Axioms of Probability | |
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| Introduction | |
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| Sample Space and Events | |
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| Axioms of Probability | |
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| Some Simple Propositions | |
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| Sample Spaces Having Equally Likely Outcomes | |
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| Probability As a Continuous Set Function | |
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| Probability As a Measure of Belief | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Conditional Probability and Independence | |
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| Introduction | |
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| Conditional Probabilities | |
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| Bayes' Formula | |
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| Independent Events | |
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| P(-[middle dot]F) is a Probability | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Random Variables | |
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| Random Variables | |
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| Discrete Random Variables | |
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| Expected Value | |
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| Expectatio of a Function of a Random Variable | |
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| Variance | |
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| The Bernoulli and Binomial Random Variables | |
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| Properties of Binomial Random Variables | |
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| Computing the Binomial Distribution Function | |
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| The Poisson Random Variable | |
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| Computing the Poisson Distribution Function | |
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| Other Discrete Probability Distribution | |
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| The Geometric Random Variable | |
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| The Negative Binomial Random Variable | |
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| The Hypergeometric Random Variable | |
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| The Zeta (or Zipf) distribution | |
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| Properties of the Cumulative Distribution Function | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Continuous Random Variables | |
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| Introduction | |
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| Expectation and Variance of Continuous Random Variables | |
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| The Uniform Random Variable | |
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| Normal Random Variables | |
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| The Normal Approximation to the Binomial Distribution | |
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| Exponential Random Variables | |
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| Hazard Rate Functions | |
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| Other Continuous Distributions | |
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| The Gamma Distribution | |
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| The Weibull Distribution | |
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| The Cauchy Distribution | |
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| The Beta Distribution | |
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| The Distribution of a Function of a Random Variable | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Jointly Distributed Random Variables | |
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| Joint Distribution Functions | |
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| Independent Random Variables | |
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| Sums of Independent Random Variables | |
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| Conditional Distributions: Discrete Case | |
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| Conditional Distributions: Continuous Case | |
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| Order Statistics | |
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| Joint Probability Distribution of Functions of Random Variables | |
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| Exchangeable Random Variables | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problem and Exercises | |
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| Properties of Expectation | |
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| Introduction | |
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| Expectation of Sums of Random Variables | |
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| Obtaining Bounds from Expectations via the Probabilistic Method | |
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| The Maximum-Minimums Identity | |
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| Covariance, Variance of Sums, and Correlations | |
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| Conditional Expectation | |
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| Definitions | |
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| Computing Expectations by Conditioning | |
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| Computing Probabilities by Conditioning | |
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| Conditional Variance | |
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| Conditional Expectation and Prediction | |
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| Moment Generating Functions | |
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| Joint Moment Generating Functions | |
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| Additional Properties of Normal Random Variables | |
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| The Multivariate Normal Distribution | |
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| The Joint Distribution of the Sample Mean and Sample Variance | |
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| General Definition of Expectation | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Limit Theorems | |
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| Introduction | |
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| Chebyshev's Inequality and the Weak Law of Large Numbers | |
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| The Central Limit Theorem | |
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| The Strong Law of Large Numbers | |
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| Other Inequalities | |
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| Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson | |
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| Summary | |
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| Problems | |
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| Theoretical Exercises | |
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| Self-Test Problems and Exercises | |
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| Additional Topics in Probability | |
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| The Poisson Process | |
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| Markov Chains | |
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| Surprise, Uncertainty, and Entropy | |
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| Coding Theory and Entropy | |
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| Summary | |
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| Theoretical Exercises and Problems | |
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| Self-Test Problems and Exercises | |
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| References | |
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| Simulation | |
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| Introduction | |
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| General Techniques for Simulating Continuous Random Variables | |
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| The Inverse Transformation Method | |
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| The Rejection Method | |
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| Simulating from Discrete Distributions | |
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| Variance Reduction Techniques | |
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| Use of Antithetic Variables | |
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| Variance Reduction by Conditioning | |
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| Control Variates | |
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| Summary | |
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| Problems | |
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| Self-Test Problems and Exercises | |
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| References | |
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| Answers to Selected Problems | |
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| Solutions to Self-Test Problems and Exercises | |
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| Index | |