
Probability Theory: A Logic of Science
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Content
- Intro
- Contents
- Preface
- Introduction
- Chapter 1
- Combinatorics
- Chapter 2
- Random Variables
- 2.1. The Simplest Probabilistic Models
- 2.2. Discrete Generalizations of the Classical Scheme
- 2.3. The General Construction of the Probability Space
- 2.4. Random Variables and Distribution Functions
- 2.5. Examples of Random Variables
- Chapter 3
- Generating and Characteristic Functions
- 3.1. Generating Functions and Their Properties
- 3.2. Characteristic Functions and Their Properties
- 3.2.1. Characteristic Functions and Polya Criterion
- 3.2.2. Polya Criterion
- Chapter 4
- Some Univariate Continuous Probability Distributions
- 4.1. Arcsine Distribution
- 4.2. Beta Distribution
- 4.3. Cauchy Distribution
- 4.4. Chi-Square Distribution
- 4.5. Exponential Distribution
- 4.6. Gamma Distribution
- 4.7. Inverse Gaussian Distribution
- 4.8. Laplace Distribution
- 4.9. Levy Distribution
- 4.10. Loglogistic Distribution
- 4.11. Logistic Distribution
- 4.12. Normal Distribution
- 4.13. Pareto Distribution
- 4.14. Power Function Distribution
- 4.15. Rayleigh Distribution
- 4.16. Stable Distribution
- 4.17. Student T-Distribution
- 4.18. Uniform Distribution
- 4.19. Weibull Distribution
- Chapter 5
- Order Statistics: From Minimum to Maximum
- 5.1. Definitions and Examples
- 5.2. Distributions of Order Statistics
- 5.3. Representations for Uniform and Exponential Order Statistics
- 5.4. Extreme Order Statistics
- Chapter 6
- Record Values and Probability Theory of Records
- 6.1. Definitions
- 6.2. Record Times and Record Values in the Case of Continuous Distributions
- 6.3. Records in the Sequences of Discrete Random Variables
- Chapter 7
- Characterizations of Continuous Distributions by Independent Copies
- Introduction
- 7.1. Arcsin Distribution
- 7.2. Beta Distribution
- 7.3. Chi-Square Distribution
- 7.4. Levy Distribution
- 7.5. Lindley Distribution
- 7.6. Loglogistic Distribution
- 7.7. Normal Distribution
- 7.8. Pearson's Random Walk
- 7.9. Power Function Distribution
- 7.10. Skew Distribution
- 7.11. Uniform Distribution
- 7.12. Von Mises distribution
- 7.13. Weibull Distribution
- 7.14. Wald Distribution
- Chapter 8
- Characterizations of Distributions by Order Statistics
- 8.1. Introduction
- 8.2. Characterizations of Distributions by Conditional Expectations
- 8.3. Characterizations by Identical Distribution
- 8.4. Characterizations by Independence Property
- Chapter 9
- Characterizations of Distributions by Record Values
- 9.1. Characterizations Using Conditional Expectations
- 9.2. Characterization by Independence Property
- 9.3. Characterizations by Identical Distribution
- Chapter 10
- Extreme Value Distributions
- Introduction
- 10.1. The PDF of Extreme Value Distributions
- 10.1.1 The Probability Density Function of Type 1 Extreme Value Distribution (T10) Is Given in Figure 10.1.1
- 10.1.2. The PDF of Type 2 Extreme Value Distributions for Xn,n
- 10.1.3. The PDF of Type 3 Distribution of Xn,n
- 10.2. Domain of Attraction
- 10.2.1. Domain of Attraction of Type I Extreme Value Distribution for Xn.n
- 10.2.2. Domain of Attraction of Type 2 Extreme Value Distribution for Xn,n
- 10.2.3. Domain of Attraction of Type 3 Extreme Value Distribution for Xn,n
- 10.3. Extreme Value Distributions for X1,n
- 10.3. Pdfs of the Extreme Value Distribution for X1,n
- 10.3.1. Type 1 Extreme Value Distribution for X1,n
- 10.3.2. Type 2 Extreme Value Distribution for X1,n
- 10.3.2. Type 3 Extreme Value Distribution for X1,n
- 10.3. Domain of Attraction of Extreme Value Distribution for X1,n
- 10.3.1. Domain of Attraction for Type 1 Extreme Value Distribution for X1,n
- 10.3.2. Domain of Attraction of Type 2 Distribution for X1,n
- 10.3.2. Domain of Attraction of Type 3 Distribution for X1,n
- 10.4. Asymptotic Distribution of the K-TH Largest Order Statistics
- Chapter 11
- Random Filling of a Segment with Unit Intervals
- 11.1. Random Filling. Continuous Case
- 11.2. Discrete Version of the Parking Problem
- Appendix
- A.1. Cauchy's Functional Equations
- A.2. Lemmas
- References
- About the Authors
- Index
- Blank Page
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