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' 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
4.6.1 Properties of Binomial Random Variables
4.6.2 Computing the Binomial Distribution Function
4.7 The Poisson Random Variable
4.7.1 Computing the Poisson Distribution Function
4.8 Other Discrete Probability Distributions
4.8.1 The Geometric Random Variable
4.8.2 The Negative Binomial Random Variable
4.8.3 The Hypergeometric Random Variable
4.8.4 The Zeta (or Zipf) Distribution
4.9 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
5.4.1 The Normal Approximation to the Binomial Distribution
5.5 Exponential Random Variables
5.5.1 Hazard Rate Functions
5.6 Other Continuous Distributions
5.6.1 The Gamma Distribution
5.6.2 The Weibull Distribution
5.6.3 The Cauchy Distribution
5.6.4 The Beta Distribution
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
7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method*
7.2.2 The Maximum-Minimums Identity*
7.3 Moments of the Number of Events that Occur
7.4 Covariance, Variance of Sums, and Correlations
7.5 Conditional Expectation
7.5.1 Definitions
7.5.2 Computing Expectations by Conditioning
7.5.3 Computing Probabilities by Conditioning
7.5.4 Conditional Variance
7.6 Conditional Expectation and Prediction
7.7 Moment Generating Functions
7.7.1 Joint Moment Generating Functions
7.8 Additional Properties of Normal Random Variables
7.8.1 The Multivariate Normal Distribution
7.8.2 The Joint Distribution of the Sample Mean and Sample Variance
7.9 General Definition of Expectation
Summary
Problems
Theoretical Exercises
Self-Test Problems and Exercises
8 Limit Theorems
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
8.6 Bounding The Error Probability
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
Theoretical Exercises
Self-Test Problems and Exercises
10 Simulation
10.1 Introduction
10.2 General Techniques for Simulating Continuous Random Variables
10.2.1 The Inverse Transformation Method
10.2.2 The Rejection Method
10.3 Simulating from Discrete Distributions
10.4 Variance Reduction Techniques
10.4.1 Use of Antithetic Variables
10.4.2 Variance Reduction by Conditioning
10.4.3 Control Variates
Summary
Problems
Self-Test Problems and Exercises
APPENDICES
A Answers to Selected Problems
B Solutions to Self-Test Problems and Exercises
Index
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