Introduction to Probability, Second Edition, discusses probability theory in a mathematically rigorous, yet accessible way. This one-semester basic probability textbook explains important concepts of probability while providing useful exercises and examples of real world applications for students to consider.
This edition demonstrates the applicability of probability to many human activities with examples and illustrations. After introducing fundamental probability concepts, the book proceeds to topics including conditional probability and independence; numerical characteristics of a random variable; special distributions; joint probability density function of two random variables and related quantities; joint moment generating function, covariance and correlation coefficient of two random variables; transformation of random variables; the Weak Law of Large Numbers; the Central Limit Theorem; and statistical inference. Each section provides relevant proofs, followed by exercises and useful hints. Answers to even-numbered exercises are given and detailed answers to all exercises are available to instructors on the book companion site.
This book will be of interest to upper level undergraduate students and graduate level students in statistics, mathematics, engineering, computer science, operations research, actuarial science, biological sciences, economics, physics, and some of the social sciences.
Key Features
- Demonstrates the applicability of probability to many human activities with examples and illustrations
- Discusses probability theory in a mathematically rigorous, yet accessible way
- Each section provides relevant proofs, and is followed by exercises and useful hints
- Answers to even-numbered exercises are provided and detailed answers to all exercises are available to instructors on the book companion site
Dedication
Preface
Overview
Chapter Descriptions
Features
Concluding Comments
Preface to the Second Edition
Chapter 1. Some Motivating Examples
Abstract
Chapter 2. Some Fundamental Concepts
Abstract
2.1 Some Fundamental Concepts
2.2 Some Fundamental Results
2.3 Random Variables
2.4 Basic Concepts and Results in Counting
Chapter 3. The Concept of Probability and Basic Results
Abstract
3.1 Definition of Probability
3.2 Some Basic Properties and Results
3.3 Distribution of a Random Variable
Chapter 4. Conditional Probability and Independence
Abstract
4.1 Conditional Probability and Related Results
4.2 Independent Events and Related Results
Chapter 5. Numerical Characteristics of a Random Variable
Abstract
5.1 Expectation, Variance, and Moment-Generating Function of a Random Variable
5.2 Some Probability Inequalities
5.3 Median and Mode of a Random Variable
Chapter 6. Some Special Distributions
Abstract
6.1 Some Special Discrete Distributions
6.2 Some Special Continuous Distributions
Chapter 7. Joint Probability Density Function of Two Random Variables and Related Quantities
Abstract
7.1 Joint d.f. and Joint p.d.f. of Two Random Variables
7.2 Marginal and Conditional p.d.f.’s, Conditional Expectation, and Variance
Chapter 8. Joint Moment-Generating Function, Covariance, and Correlation Coefficient of Two Random Variables
Abstract
8.1 The Joint m.g.f. of Two Random Variables
8.2 Covariance and Correlation Coefficient of Two Random Variables
8.3 Proof of Theorem 1, Some Further Results
Chapter 9. Some Generalizations to k Random Variables, and Three Multivariate Distributions
Abstract
9.1 Joint Distribution of k Random Variables and Related Quantities
9.2 Multinomial Distribution
9.3 Bivariate Normal Distribution
9.4 Multivariate Normal Distribution
Chapter 10. Independence of Random Variables and Some Applications
Abstract
10.1 Independence of Random Variables and Criteria of Independence
10.2 The Reproductive Property of Certain Distributions
10.3 Distribution of the Sample Variance under Normality
Chapter 11. Transformation of Random Variables
Abstract
11.1 Transforming a Single Random Variable
11.2 Transforming Two or More Random Variables
11.3 Linear Transformations
11.4 The Probability Integral Transform
11.5 Order Statistics
Chapter 12. Two Modes of Convergence, the Weak Law of Large Numbers, the Central Limit Theorem, and Further Results
Abstract
12.1 Convergence in Distribution and in Probability
12.2 The Weak Law of Large Numbers and the Central Limit Theorem
12.3 Further Limit Theorems
Chapter 13. An Overview of Statistical Inference
Abstract
13.1 The Basics of Point Estimation
13.2 The Basics of Interval Estimation
13.3 The Basics of Testing Hypotheses
13.4 The Basics of Regression Analysis
13.5 The Basics of Analysis of Variance
13.6 The Basics of Nonparametric Inference
Appendix 20. Appendix
Chapter 21. Some Notation and Abbreviations
Appendix 22. Answers to Even-Numbered Exercises
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Appendix 23. Revised Answers Manual to Introduction to Probability
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Index
