1. COMBINATORIAL ANALYSIS
1.1 Introduction
1.2 The Basic Principle of Counting
1.3 Permutations
1.4 Combinations
1.5 Multinomial Coefficients
1.6 The Number of Integer Solutions of Equations
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
2. AXIOMS OF PROBABILITY
2.1 Introduction
2.2 Sample Space and Events
2.3 Axioms of Probability
2.4 Some Simple Propositions
2.5 Sample Spaces Having Equally Likely Outcomes
2.6 Probability as a Continuous Set Function
2.7 Probability as a Measure of Belief
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
3. CONDITIONAL PROBABILITY AND INDEPENDENCE
3.1 Introduction
3.2 Conditional Probabilities
3.3 Bayes’s Formula
3.4 Independent Events
3.5 P(·|F) Is a Probability
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
4. RANDOM VARIABLES
4.1 Random Variables
4.2 Discrete Random Variables
4.3 Expected Value
4.4 Expectation of a Function of a Random Variable
4.5 Variance
4.6 The Bernoulli and Binomial Random Variables
4.7 The Poisson Random Variable
4.8 Other Discrete Probability Distributions
4.9 Expected Value of Sums of Random Variables
4.10 Properties of the Cumulative Distribution Function
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
5. CONTINUOUS RANDOM VARIABLES
5.1 Introduction
5.2 Expectation and Variance of Continuous Random Variables
5.3 The Uniform Random Variable
5.4 Normal Random Variables
5.5 Exponential Random Variables
5.6 Other Continuous Distributions
5.7 The Distribution of a Function of a Random Variable
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
6. JOINTLY DISTRIBUTED RANDOM VARIABLES
6.1 Joint Distribution Functions
6.2 Independent Random Variables
6.3 Sums of Independent Random Variables
6.4 Conditional Distributions: Discrete Case
6.5 Conditional Distributions: Continuous Case
6.6 Order Statistics
6.7 Joint Probability Distribution of Functions of Random Variables
6.8 Exchangeable Random Variables
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
7. PROPERTIES OF EXPECTATION
7.1 Introduction
7.2 Expectation of Sums of Random Variables
7.3 Moments of the Number of Events that Occur
7.4 Covariance, Variance of Sums, and Correlations
7.5 Conditional Expectation
7.6 Conditional Expectation and Prediction
7.7 Moment Generating Functions
7.8 Additional Properties of Normal Random Variables
7.9 General Definition of Expectation
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
8. LIMIT THEOREMS 394
8.1 Introduction
8.2 Chebyshev’s Inequality and the Weak Law of Large Numbers
8.3 The Central Limit Theorem
8.4 The Strong Law of Large Numbers
8.5 Other Inequalities and a Poisson Limit Result
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable
8.7 The Lorenz Curve
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
9. ADDITIONAL TOPICS IN PROBABILITY
9.1 The Poisson Process
9.2 Markov Chains
9.3 Surprise, Uncertainty, and Entropy
9.4 Coding Theory and Entropy
Summary
Problems and Theoretical Exercises
Self-Test Problems and Exercises
10. SIMULATION
10.1 Introduction
10.2 General Techniques for Simulating Continuous Random Variables
10.3 Simulating from Discrete Distributions
10.4 Variance Reduction Techniques
Summary
Problems
Self-Test Problems and Exercises
Answers to Selected Problems
Solutions to Self-Test Problems and Exercises
Index
