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▶ 이 책은 Essentials of Probability for Statistics를 다룬 이론서입니다. Essentials of Probability for Statistics의 기초적이고 전반적인 내용을 학습할 수 있습니다.

I Basic Probability

1 What is probability?
1.1 Sample spaces and events
1.2 Probability measure
1.3 Counting methods
1.4 Probability of a Union
1.5 Exercises

2 Conditional Probability
2.1 Conditional probability
2.2 Law of total probability
2.3 Bayes’ theorem
2.4 Independent events
2.5 Exercises

3 Discrete random variables
3.1 Discrete random variables
3.2 Moments
3.3 Joint distributions
3.4 Waiting time distributions
3.5 Poisson distribution
3.6 Exercises

II Continuous Distributions

4 Continuous random variables
4.1 Continuous random variables
4.2 Moments
4.3 Normal distribution
4.4 Waiting time distributions
4.5 Beta distributions
4.6 Exercises

5 Joint distributions
5.1 Bivariate random variables
5.2 Independent random variables
5.3 Transformation of two random variables
5.4 Joint moments
5.5 Exercises

6 Conditional distributions
6.1 Conditional density
6.2 Conditional expectation
6.3 Conditional variance
6.4 Bivariate normal distributions
6.5 Multivariate distributions
6.6 Exercises

'This book is designed for a first course in probability at an undergraduate level, targeting students who have studied calculus at the university level and are seeking an introduction to probability with mathematical content. This book covers all of the standard material for such courses, but it also contains many topics that are usually not found in introductory probability books ? such as simulation using R and additional material useful for mathematical statistics. Where possible, I tried to provide mathematical details, and it is expected that students are seeking to gain some mastery over these. For use in a standard one-semester course, in which both discrete and continuous probability is covered, students should have taken a prerequisite of calculus, including an introduction to multiple integrals, for example, STAT 201 provided by the department of statistics in Korea University. The main mathematical challenge lies not in performing technical calculus derivations, but in translating between abstract concepts and concrete examples. This book is also intended to prepare students for advanced work in probability, who are intending to go on to take mathematical statistics, stochastic processes, computational methods, or any other advanced probability course.

The emphasis throughout the book is on probability, but attention is also given to statistics. The book can be used in a variety of course lengths, levels and areas of emphasis, and is suitable not only for introductory courses, but also for self-study. It presents a comprehensive treatment of probability ideas and techniques essential for a firm understanding of probability and statistics. Chapter 1 introduces the concept of probability and covers sample spaces and events, the axioms of probability, and counting. The basic properties of a probability measure are also developed. Chapter 2 deals with conditional probability, law of total probability, Bayes’ theorem, and independence of events. Discrete random variables and their distributions are the subject of Chapter 3. Chapter 4 introduces continuous random variables and their distributions. Joint continuous distributions are the focus of Chapter 5, including marginal distributions, expectations of a function of random variables, and the effects of change of variables. Chapter 6 mainly discusses conditional distributions, including conditional expectations and variances. The last two sections in Chapter 6 deal with bivariate normal distribution theory and multivariate distributions as advanced topics. All the chapters include material on Monte Carlo methods using a statistical software, R. Simulation is a key aspect of the application of probability theory, and it is my view that its teaching can be integrated with the theory from the start. This reveals the power of probability to solve real-world problems and helps convince students that it is far more than just an interesting mathematical result. This material can be skipped, if a student or an instructor believes otherwise or feels there is not enough time to cover it effectively.

The book contains more than the usual number of examples worked out in detail, ranging from straightforward to reasonably challenging. It is not possible to go through all these examples in class. Rather, a student is expected to deal quickly with the main points of theory, then spend working time on examples and exercises from the book. A student cannot hope to really learn the material simply by sitting passively in the classroom and listening to the instructor. The student must get actively involved in working problems.
I am indebted to my students without whom I could not have written this book. They helped me with writing and revising earlier versions of this text. In particular, Eunji Lee spent many, many hours editing LATEX code for the book, which is a modified version of the Legrand Orange Book LATEX Template Version 2.1.1.The best aspect of the book’s layout are due to her hard work; any stylistic deficiencies are due to my lack of expertise. Sunsik Kim also worked hard to edit and produce many figures together with Eunji Lee. Dongu Han and Hyunsue Jung read through the entire book very carefully and gave me detailed comments for improvements and corrections. Also, I would like to thank all the students in STAT 221 for valuable comments, suggestions, and inspiration.

Finally, you should note that the book contains a random number of errors that may follow Poisson distributions. Errors unavoidably arise in any project like this (meaning a project in which I am involved). For this reason, I will try to post information about the book, including a list of corrections, on the available link of the publisher’s website. Readers are encouraged to inform me of any errors that they discover.