(NOTE: Each chapter concludes with Summary, Problems, Theoretical Exercises, and Self-Test Problems and Exercises.)
1. Combinatorial Analysis.
Introduction. The Basic Principle of Counting. Permutations. Combinations. Multinomial Coefficients. The Number of Integer Solutions of Equations.
2. Axioms of Probability.
Introduction. Sample Space and Events. Axioms of Probability. Some Simple Propositions. Sample Spaces Having Equally Likely Outcomes. Probability As a Continuous Set Function. Probability As a Measure of Belief.
3. Conditional Probability and Independence.
Introduction. Conditional Probabilities. Bayes' Formula. Independent Events. P(*|F) is a Probability.
4. Random Variables.
Random Variables. Discrete Random Variables. Expected Value. Expectation of a Function of a Random Variable. Variance. The Bernoulli and Binomial Random Variables. The Poisson Random Variable. Other Discrete Probability Distribution. Properties of the Cumulative Distribution Function.
5. Continuous Random Variables.
Introduction. Expectation and Variance of Continuous Random Variables. The Uniform Random Variable. Normal Random Variables. Exponential Random Variables. Other Continuous Distributions. The Distribution of a Function of a Random Variable.
6. Jointly Distributed Random Variables.
Joint Distribution Functions. Independent Random Variables. Sums of Independent Random Variables. Conditional Distributions: Discrete Case. Conditional Distributions: Continuous Case. Order Statistics. Joint Probability Distribution of Functions of Random Variables. Exchangeable Random Variables.
7. Properties of Expectation.
Introduction. Expectation of Sums of Random Variables. Covariance, Variance of Sums, and Correlations. Conditional Expectation. Conditional Expectation and Prediction. Moment Generating Functions. Additional Properties of Normal Random Variables. General Definition of Expectation.
8. Limit Theorems.
Introduction. Chebyshev's Inequality and the Weak Law of Large Numbers. The Central Limit Theorem. The Strong Law of Large Numbers. Other Inequalities. Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson.
9. Additional Topics in Probability.
The Poisson Process. Markov Chains. Surprise, Uncertainty, and Entropy. Coding Theory and Entropy.
10. Simulation.
Introduction. General Techniques for Simulating Continuous Random Variables. Simulating from Discrete Distributions. Variance Reduction Techniques.
Appendix A. Answers to Selected Problems.
Appendix B. Solutions to Self-Test Problems and Exercises.
Index.
