Probability Essentials

As far as beginning graduate-level books on probability are concerned this is definitely one of the best. This looks like a set of lectures turned into a book.The competition in my mind would beFirst Look at Rigorous Probability Theory (more compact, perhaps a little dense)Probability & Measure Theory, Second Edition (covers more ground and is very clear)A Course in Probability Theory, Revised Edition, Second Edition(very detailed explanations, but you should probably have followed a course on measure theory)Please dont dive into probability at this level, your intuition might not be ready for it.To do that I recommendAn Introduction to Probability Theory and Its Applications, Vol. 1 (Volume 1) (if you have the time)There is also Basic Probability Theory (Dover Books on Mathematics) which is an excellent introduction stopping short of measure theory.

Very good. Awesome.

Good

the statistics lecturer at my university recommended this book, and they choose to purchase it for my studies. The book holds a lot of information, and is definitely worthwhile for those who are interested in numbers. Formulas are well explained and properly documented through methodical examples.

This is probably my favorite math book so far. It is written in a format which I haven't found in any other book I read: It has very short chapters and uses uniquely concise notation. The latter is especially appealing if like me one isn't a fan of heavily analytical notation. In this book, probability theory comes first, measure theory second, and the notation reflects this: For example expectation is used instead of integrals whenever possible.The book has only minor faults, like the rare lapse in rigor, or incomprehensible step in a proof. Since this won't be especially harmful to a reader with sufficient mathematical maturity, and because the notational style in my opinion provides such a great benefit, I rate this book 4.5 stars with a tendency towards 5.I would recommend this book for undergraduate students in their final semesters with prior exposure to non-measure-theoretic probability theory (Europe), and for beginning graduate students in the US.

This book is my favorite to use as a basis for an introduction to probability theory course. It’s graduate-level material designed to give a rigorous basis for later probability based courses, and it succeeds admirably. The presentation is clear, detailed, and structured in a way that it is fairly easy to design a course around the book—one can make it a required textbook or just use it as a guide to form lecture notes. The same clear exposition and sufficient detail enables it to be used as a self-study guide, too, as long as one has a reasonably rigorous background in mathematics.It’s a bit slim and lecture/lesson-oriented to be that great as a reference book, but it’s great to build a course around, and not overly expensive for students. Highest recommendation.

Terse exposition, but while this usually helps results stand on their own and encourages the reader to fill in the details, the book's page layout makes the material look chaotic and un-clear. Very little motivation or intuition is included as part of the anemic chapter length.Endlessly frustrating as a reference. Table of contents gives absolutely no fine information about where topics are located, and the index is as terse as the rest of the book. Good luck finding anything efficiently. Many times when I try to work with this book I've had to waste 10+ minutes flipping around to check basic definitions - probability density, state space, expectation, variance.Exercises are semi interesting, but the authors decided it was a good idea to include answers to some of them. In my opinion, this is idiotic in a graduate level book, as at this point students should be able to work independently or know to ask their peers/teachers for help if they are stuck. Even worse is the number of times these answers are just plain wrong. Were the authors purposely trying to deceive students in an effort to 'help' them learn? This is absurd and unnecessary. This degrades this book from a 'textbook' to a grade school style 'workbook'Its only redeeming value is that it is cheap, but I can think of cheaper ways to decorate my desk. If you have the option, choose a different option.

The opening chapters (1-6) provide a decent and readable introduction to key concepts in measure theory: sigma-algebras, (probability) measures, random variables, etc. However, the middle and later chapters are written like lecture notes --definition, theorem, proof; theorem proof; theorem, proof, corollary -- with little motivation or explanation of relevance to measure theoretic probability, i.e. the lecturer would provide such motivations and explanations (unfortunately the book does not come with a lecturer). The chapters on martingales are thorough--but read like a reference rather than a text-- and the token chapter on the Radon-Nikodym theorem fails to capture its importance in measure theoretic probability. Overall, this book serves as a decent introduction, but I would recommend supplementing the material with corresponding material from e.g. Ash's Probability and Measure Theory or Billingsley's Probability and Measure.