Probability Theory: A Logic of Science

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
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