INTRODUCTION TO PROBABILITY THEORY
with contemporary applications
CONTENTS
Preface
1. Classical Probability
1.1 Beginnings
1.2 Basic Rules
1.3 Counting
1.4 Equally Likely Case
1.5 Other Models
2. Axioms of Probability
2.1 Introduction
2.2 Set Theory
2.3 Countable Sets
2.4 Axioms
2.5 Properties of Probability Functions
2.6 Conditional Probability and Independence
2.7 Some Applications
3. Random Variables
3.1 Introduction
3.2 Random Variables
3.3 Independent Random Variables
3.4 Generating Functions
3.5 Gambler's Ruin Problem
3.6 Appendix
4. Expectation
4.1 Introduction
4.2 Expected Value
4.3 Properties of Expectation
4.4 Covariance and Correlation
4.5 Conditional Expectation
4.6 Entropy
5. Stochastic Processes
5.1 Introduction
5.2 Markov Chains
5.3 Random Walks
5.4 Branching Processes
5.5 Prediction Theory
6. Continuous Random Variables
6.1 Introduction
6.2 Random Variables
6.3 Distribution Functions
6.4 Joint Distribution Functions
6.5 Computations with Densities
6.6 Multivariate and Conditional Densities
7. Expectation Revisited
7.1 Introduction
7.2 Riemann-Stieltjes Integral
7.3 Expectation and Conditional Expectation
7.4 Normal Density
7.5 Covariance and Covariance Functions
8. Continuous Parameter Markov Processes
8.1 Introduction
8.2 Poisson Process
8.3 Birth and Death Processes
8.4 Markov Chains
8.5 Matrix Calculus
8.6 Stationary Distributions
Solutions to Exercises
Standard Normal Distribution Function
Symbols
Index