An introduction to probability theory and its applications

What Hacker News says

1. mamonsterMar 2024
   William Feller: An introduction to Probability Theory and Its Applications Volumes 1 and 2 - Volume 1 is considered the "classic" in most math undergrad courses, Volume 2 is for masters level students.

   Billingsley Probability and Measure - For more advanced probability you need to know measure theory, this is a book dedicated to that.

   Karlin + Taylor - First (and Second) Course in Stochastic Processes - If you are interested in learning Ito calculus/stochastics those 2 books are pretty good.

   These books have some overlap between themselves but are pretty complementary overall. I would say Feller Volume 1 up to Chapter 11 is decent for non-math people, beyond that you need to be serious about learning.

   Didn't put any statistics books as that's a whole different topic.

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2. antegamisouJun 2023
   Anything that would help build a strong probability theory background.

   Assuming you have some freshman-college math literacy, here are a few great introductory books to Probability theory:

       Introduction To Probability by D. Bertsekas & J. Tsitsiklis
   

   https://www.amazon.com/Introduction-Probability-2nd-Dimitri-...

       Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance  by K. L. Chung & F. AitSahlia 
   

   https://www.amazon.com/Elementary-Probability-Theory-Introdu...

       An Introduction to Probability Theory and Its Applications Vol. 1 by W. Feller
   

   https://www.amazon.com/Introduction-Probability-Theory-Appli...

   I'm sure there are some great video lectures on the subject as well, but unfortunately I can't point you to any relevant material since textbooks is what I used when I was in college. But if I had to guess, the latest relevant MIT OCW course that has video lectures available would be sufficient.

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3. lajosbacsNov 2022
   An Introduction to Probability Theory and its Applications

   By William Feller, written in 1968 but feels very fresh even today.

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4. francassoMar 2018
   My own take on it:

   Lang's books about the basics I find lovely

   - Basic stuff: Basic Mathematics - Lang

   - One variable Calculus: A First Course in Calculus - Lang

   - Multi variable Calculus: Calculus of Several Variables - Lang

   - Linear Algebra: Linear Algebra - Lang

   - Number theory: An Introduction to the Theory of Numbers - Niven, Zuckerman, Montgomery

   Some more advanced stuff

   - Algebra: Algebra - Artin or Algebra, Chapter 0 - Aluffi

   - Complex Analysis: Functions of One Complex Variable I - Conway

   - Probability Basics: An Introduction to Probability Theory and its Applications I - Feller

   - Real Analysis and functional analysis basics: Real Analysis - Folland

   - Basic Differential Geometry: Elementary Differential Geometry - O'Neill

   - Riemann Surfaces (algebraic take): Algebraic Curves - Fulton

   - Differential Topology: Differential Topology - Guillemin, Pollack

   - Riemann Surfaces (analytic take): Compact Riemann Surfaces, an Introduction to Contemporary Mathematics - Jost

   - Modern Differential Geometry: Lectures on The Geometry of Manifolds - Nicolaescu

   - Functional analysis: Fundamentals of the Theory of Operator Algebras I - Kadison, Ringrose

   - Introduction to the Index Theory of which you actually have already seen some in the Riemann Surfaces books with the Riemann-Roch theorem: Index Theory with Applications to Mathematics and Physics - Bleecker, Doob Bavnbek

   - Homological Algebra: An Introduction to Homological Algebra - Weibel

   - Algebra for algebraic geometry: Commutative Ring Theory - Matsumura

   - Soft introduction to schemes: The Geometry of Schemes - Eisenbud, Harris

   - Algebraic geometry: Algebraic Geometry - Robin Hartshorne

   This should get you up more or less to what was current in the '60s :)

   Additional methodological note: I'm not suggesting going linearly through all these books. Well, perhaps going linearly thorough the basics would be a good idea, but after that I would follow my own interests.

   The important thing is really to have pen and paper and work things out by yourself, not just reading the book. I'm not saying you should try to prove all the theorems yourself or do all the exercises, that would take an unrealistic amount of time, but you can try to think about a theorem before reading its proof to see if you have a sense of which road is more likely to lead to a proof, and then try to reproduce the proof with pen and paper after having read it to check if you actually understood it.

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5. MyrmornisMar 2014
   Well it's too late to get any points for this inference now, but I'm going to claim that there was a strong clue that the author of the PDF was old: the bitcoin paper cites "An Introduction to Probability Theory and Its Applications" by William Feller. This is a classic, from the 1960s, but I don't think it's very well known among people under 40 (correct me if I'm wrong).
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6. waldrewsOct 2009
   Sounds like you're more into probability modelling and machine learning than statistics in the traditional hypothesis testing sense. Besides ESL, a book I'd recommend is Bishop's Pattern Recognition and Machine Learning. It starts from the beginning of probability theory applied to computer science problems, and covers every modern topic.

   videolectures.net is filled with lectures on CS-flavored probability modelling and machine learning topics. The best bet is the multi-hour "tutorial" lecture series and minicourses; it may take a while to choose the right starting point.

   For serious stats and probability without the CS flavoring (not useful for the quick-road-to-hacking-power agenda):

   For classical deep stats theory, everyone I know begins with Cassela and Berger's Statistical Inference. Don't expect algorithms in this though.

   On the probability side: Feller's An Introduction to Probability Theory and Its Applications. Deep, readable, sometimes funny, full of "whoa" insights. Would be hard to actually grok every chapter in both volumes, but you read it for insight into the power of probability and then use it as a reference.

   (grad student in stats, among other things)

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