An Introduction to Probability. With MATHEMATICA

✍ Scribed by Kao Edward

• Publisher: World Scientific Publishing
• Year: 2022
• Tongue: English
• Leaves: 390
• Category: Library

✦ Table of Contents

Contents
Preface
Notations and Abbreviations
Chapter 1 Permutation and Combination
1.1. Introduction
1.2. Permutations
1.3. Combinations
1.4. Basic Principles of Counting
1.5. Binomial and Multinomial Coefficients
1.6. Occupancy Problems
1.7. Combinatorial Generating Functions
1.8. Exploring with Mathematica
Problems
Remarks and References
Chapter 2 Axioms of Probability
2.1. Introduction
2.2. Sample Space, Events, and Sets
2.3. Axioms of Probability
2.4. Sample Space Having Equally Likely Outcomes
2.5. The Importance of Being Truly Random
2.6. Probability as a Continuous Set Function
2.7. The Subjective Probability
2.8. Exploring with Mathematica
Problems
Remarks and References
Chapter 3 Conditional Probability
3.1. Introduction
3.2. Conditional Probabilities and Independence
3.3. The Law of Total Probabilities
3.4. Bayes Theorem
3.5. Artificial Intelligence and Bayes Theorem
3.6. More on Problem Solving by Conditioning
3.7. Exploring with Mathematica
Problems
Remarks and References
Chapter 4 Random Variables
4.1. Introduction
4.2. Distribution Functions
4.3. Functions of a Random Variable
4.4. Expectation of a Random Variable
4.5. Random Variables that are Neither Discrete nor Continuous
4.6. Various Transforms for Applications in Probability
4.7. Higher Moments of a Random Variable
4.8. Exploring with Mathematica
Problems
Remarks and References
Chapter 5 Discrete Random Variables
5.1. Introduction
5.2. Bernoulli and Binomial Random Variables
5.3. Hypergeometric Random Variables
5.4. Poisson Random Variables
5.5. Geometric and Negative Binomial Random Variables
5.6. Probability Generating Functions
5.7. Summary
5.8. Exploring with Mathematica
Problems
Remarks and References
Chapter 6 Continuous Random Variables
6.1. Introduction and Transformation of Random Variables
6.2. Laplace Transforms and Characteristic Functions
6.3. Uniform and Exponential Random Variable
6.4. Erlang and Gamma Random Variables
6.5. Weibull Random Variables
6.6. Normal and Lognormal Random Variables
6.7. More Continuous Random Variables and Variance Gamma
6.8. Summary
6.9. Exploring with Mathematica
Problems
Remarks and References
Chapter 7 Jointly Distributed Random Variables
7.1. Introduction and Distribution Functions
7.2. Independent Random Variables
7.3. Order Statistics
7.4. Conditional Random Variables
7.5. Functions of Jointly Distributed Random Variables
7.6. More Well-Known Distributions
7.7. Mixtures of Random Variables
7.8. Exploring with Mathematica
Problems
Remarks and References
Chapter 8 Dependence and More on Expectations
8.1. Introduction
8.2. Exchangeable Random Variables
8.3. Dependence
8.4. Bivariate Normal Distributions
8.5. More on Expectations and Related Subjects
8.6. Sampling from a Finite Population
8.7. Exploring with Mathematica
Problems
Remarks and References
Chapter 9 Limit Theorems
9.1. Markov and Chebyshev’s Inequalities
9.2. Various Forms of Convergence
9.3. Characteristics Functions
9.4. Weak Law of and Strong Law of Large Numbers
9.5. Central Limit Theorem
9.6. Other Inequalities
9.7. Exploring with Mathematica
Problems
Remarks and References
A Terse Introduction to Mathematica
Four Simple Examples
Answers to Odd-Numbered Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Index