Probability Theory: An Introduction Using R

✍ Scribed by Shailaja R. Deshmukh, Akanksha S. Kashikar

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 596
• Edition: 1
• Category: Library

✦ Synopsis

This book introduces Probability Theory with R software and explains abstract concepts in a simple and easy-to-understand way by combining theory and computation. It discusses conceptual and computational examples in detail, to provide a thorough understanding of basic techniques and develop an enjoyable read for students seeking suitable material for self-study. It illustrates fundamental concepts including fields, sigma-fields, random variables and their expectations, various modes of convergence of a sequence of random variables, laws of large numbers and the central limit theorem.

• Computational exercises based on R software are included in each Chapter

• Includes a brief introduction to the basic functions of R software for beginners in R and serves as a ready reference

• Includes Numerical computations, simulation studies, and visualizations using R software as easy tools to explain abstract concepts

• Provides multiple-choice questions for practice

• Incorporates self-explanatory R codes in every chapter

This textbook is for advanced students, professionals, and academic researchers of Statistics, Biostatistics, Economics and Mathematics.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
List of Tables
List of Figures
Preface
Author Biographies
1. Sigma Field, Borel Field and Probability Measure
1.1. Introduction
1.2. Introduction to R Software and Language
1.3. Limit of a Sequence of Sets
1.4. Borel Field
1.5. Probability Space
1.6. Conceptual Exercises
1.7. Computational Exercises
1.8. Multiple Choice Questions
2. Random Variables and Random Vectors
2.1. Introduction
2.2. Random Variables
2.3. Random Vectors
2.4. Simple Random Variable
2.5. Probability Distribution of a Random Variable
2.6. Conceptual Exercises
2.7. Computational Exercises
2.8. Multiple Choice Questions
3. Distribution Function
3.1. Introduction
3.2. Properties of a Distribution Function
3.3. Decomposition of a Distribution Function
3.4. Distribution Function of a Random Vector
3.5. Conceptual Exercises
3.6. Computational Exercises
3.7. Multiple Choice Questions
4. Expectation and Characteristic Function
4.1. Introduction
4.2. Expectation of a Simple Random Variable
4.3. Expectation of Non-Negative and Arbitrary Random Variables
4.4. Characteristic Function
4.5. Moment Inequalities
4.6. Conceptual Exercises
4.7. Multiple Choice Questions
5. Independence
5.1. Introduction
5.2. Independence of Events and Classes of Events
5.3. Independence of Random Variables
5.4. Kolmogorov Zero-One Law
5.5. Conceptual Exercises
5.6. Multiple Choice Questions
6. Almost Sure Convergence and Borel Zero-One Law
6.1. Introduction
6.2. Definitions of Various Modes of Convergence
6.3. Almost Sure Convergence
6.4. Borel Zero-One Law
6.5. Conceptual Exercises
6.6. Computational Exercises
6.7. Multiple Choice Questions
7. Convergence in Probability, in Law and in r-th Mean
7.1. Introduction
7.2. Convergence in Probability
7.3. Convergence in Law
7.4. Convergence in r-th Mean
7.5. Conceptual Exercises
7.6. Computational Exercises
7.7. Multiple Choice Questions
8. Convergence of a Sequence of Expectations
8.1. Introduction
8.2. Monotone Convergence Theorem
8.3. Lebesgue Dominated Probability Convergence Theorem
8.4. Conceptual Exercises
8.5. Computational Exercises
8.6. Multiple Choice Questions
9. Laws of Large Numbers
9.1. Introduction
9.2. Weak Law of Large Numbers
9.3. Strong Law of Large Numbers
9.4. Conceptual Exercises
9.5. Computational Exercises
9.6. Multiple Choice Questions
10. Central Limit Theorem
10.1. Introduction
10.2. Lindeberg-Levy CLT
10.3. CLT for Independent Random Variables
10.4. Conceptual Exercises
10.5. Computational Exercises
10.6. Multiple Choice Questions
11. Solutions to Conceptual Exercises
11.1. Chapter 1
11.2. Chapter 2
11.3. Chapter 3
11.4. Chapter 4
11.5. Chapter 5
11.6. Chapter 6
11.7. Chapter 7
11.8. Chapter 8
11.9. Chapter 9
11.10. Chapter 10
Bibliography
Index