Preface |
Combinatorial Analysis / 1: |
Introduction / 1.1: |
The Basic Principle of Counting / 1.2: |
Permutations / 1.3: |
Combinations / 1.4: |
Multinomial Coefficients / 1.5: |
The Number of Integer Solutions of Equations / 1.6: |
Summary |
Problems |
Theoretical Exercises |
Self-Test Problems and Exercises |
Axioms of Probability / 2: |
Sample Space and Events / 2.1: |
Some Simple Propositions / 2.3: |
Sample Spaces Having Equally Likely Outcomes / 2.5: |
Probability As a Continuous Set Function / 2.6: |
Probability As a Measure of Belief / 2.7: |
Conditional Probability and Independence / 3: |
Conditional Probabilities / 3.1: |
Bayes' Formula / 3.3: |
Independent Events / 3.4: |
P(-[middle dot]F) is a Probability / 3.5: |
Random Variables / 4: |
Discrete Random Variables / 4.1: |
Expected Value / 4.3: |
Expectatio of a Function of a Random Variable / 4.4: |
Variance / 4.5: |
The Bernoulli and Binomial Random Variables / 4.6: |
Properties of Binomial Random Variables / 4.6.1: |
Computing the Binomial Distribution Function / 4.6.2: |
The Poisson Random Variable / 4.7: |
Computing the Poisson Distribution Function / 4.7.1: |
Other Discrete Probability Distribution / 4.8: |
The Geometric Random Variable / 4.8.1: |
The Negative Binomial Random Variable / 4.8.2: |
The Hypergeometric Random Variable / 4.8.3: |
The Zeta (or Zipf) distribution / 4.8.4: |
Properties of the Cumulative Distribution Function / 4.9: |
Continuous Random Variables / 5: |
Expectation and Variance of Continuous Random Variables / 5.1: |
The Uniform Random Variable / 5.3: |
Normal Random Variables / 5.4: |
The Normal Approximation to the Binomial Distribution / 5.4.1: |
Exponential Random Variables / 5.5: |
Hazard Rate Functions / 5.5.1: |
Other Continuous Distributions / 5.6: |
The Gamma Distribution / 5.6.1: |
The Weibull Distribution / 5.6.2: |
The Cauchy Distribution / 5.6.3: |
The Beta Distribution / 5.6.4: |
The Distribution of a Function of a Random Variable / 5.7: |
Jointly Distributed Random Variables / 6: |
Joint Distribution Functions / 6.1: |
Independent Random Variables / 6.2: |
Sums of Independent Random Variables / 6.3: |
Conditional Distributions: Discrete Case / 6.4: |
Conditional Distributions: Continuous Case / 6.5: |
Order Statistics / 6.6: |
Joint Probability Distribution of Functions of Random Variables / 6.7: |
Exchangeable Random Variables / 6.8: |
Self-Test Problem and Exercises |
Properties of Expectation / 7: |
Expectation of Sums of Random Variables / 7.1: |
Obtaining Bounds from Expectations via the Probabilistic Method / 7.2.1: |
The Maximum-Minimums Identity / 7.2.2: |
Covariance, Variance of Sums, and Correlations / 7.3: |
Conditional Expectation / 7.4: |
Definitions / 7.4.1: |
Computing Expectations by Conditioning / 7.4.2: |
Computing Probabilities by Conditioning / 7.4.3: |
Conditional Variance / 7.4.4: |
Conditional Expectation and Prediction / 7.5: |
Moment Generating Functions / 7.6: |
Joint Moment Generating Functions / 7.6.1: |
Additional Properties of Normal Random Variables / 7.7: |
The Multivariate Normal Distribution / 7.7.1: |
The Joint Distribution of the Sample Mean and Sample Variance / 7.7.2: |
General Definition of Expectation / 7.8: |
Limit Theorems / 8: |
Chebyshev's Inequality and the Weak Law of Large Numbers / 8.1: |
The Central Limit Theorem / 8.3: |
The Strong Law of Large Numbers / 8.4: |
Other Inequalities / 8.5: |
Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson / 8.6: |
Additional Topics in Probability / 9: |
The Poisson Process / 9.1: |
Markov Chains / 9.2: |
Surprise, Uncertainty, and Entropy / 9.3: |
Coding Theory and Entropy / 9.4: |
Theoretical Exercises and Problems |
References |
Simulation / 10: |
General Techniques for Simulating Continuous Random Variables / 10.1: |
The Inverse Transformation Method / 10.2.1: |
The Rejection Method / 10.2.2: |
Simulating from Discrete Distributions / 10.3: |
Variance Reduction Techniques / 10.4: |
Use of Antithetic Variables / 10.4.1: |
Variance Reduction by Conditioning / 10.4.2: |
Control Variates / 10.4.3: |
Answers to Selected Problems / Appendix A: |
Solutions to Self-Test Problems and Exercises / Appendix B: |
Index |
Preface |
Combinatorial Analysis / 1: |
Introduction / 1.1: |