Linde W. Probability Theory: A First Course in Probability Theory and Statistics

2nd. ed. - Berlin: Walter de Gruyter, 2024. - 500 p. - (De Gruyter Textbook). - ISBN: 3111324842.This book is intended as an introduction to Probability Theory and Mathematical Statistics for students in mathematics, the physical sciences, engineering, and related fields. It is based on the author’s 25 years of experience teaching probability and is aimed at helping students overcome common difficulties in learning the subject. The focus of the book is an explanation of the theory, mainly by the use of many examples. Whenever possible, proofs of stated results are provided. All sections conclude with a short list of problems. The book also includes several optional sections on more advanced topics. This textbook would be ideal for use in a first course in Probability Theory.Changes in the second edition:There are a few changes to the second edition. For instance, I added more than 40 new examples, so now the book contains about 280 of them. Among the new ones are some classical examples, e.g., the “Boy or Girl Paradox,” the “Secretary Problem,” the “Two-Envelope Paradox,” or “Gambler’s Ruin,”. Other examples have been included for a better understanding of the general, sometimes quite abstract, topics. The section about Mathematical Statistics contains now three tables that summarize the main tests and confidence regions for normally distributed populations. I hope that these résumés help to get a quick overview of the most used techniques in Mathematical Statistics. Furthermore, there is a new Section about confidence regions for hypergeometric distributed samples, an important topic missing in the first edition. A completely new ingredient in this edition is short summaries at the end of almost every section. Here I give a compressed overview about basic notions and results presented in the preceding section. These abstracts aim to tell the reader what were the most important statements and what can be omitted from the first reading. I believe that graphical presentations of abstract mathematical statements are a very helpful aid for better understanding, not only for beginners. Therefore, I added more than 80 new figures, so that now the book contains more than 100 of them. Some minor misprints or incorrect arguments have been eliminated, a few parts were rewritten to make them, as I hope, clearer and better understandable. Besides, I added several new problems, updated the list of references, and completed it by adding a few classical books about Measure Theory and Probability.Probabilities.
Conditional Probabilities and Independence.
Random Variables and Their Distribution.
Operations on Random Variables.
Expected Value, Variance, and Covariance.
Normally Distributed Random Vectors.
Limit Theorems.
Mathematical Statistics.True PDF