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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 | |