26
Jun 15
  1. Probability,Statistics And Stochastic Processes (Undergraduate Level; pre-measure theory)

 

Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave nether room nor demand for a theory of probabilities.-George Boole

 

 

Probability is a mathematical discipline whose aims are akins to those, for example, of geometry of analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, and (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation.-William Feller

 

 

  Undergraduate Stochastic Processes Richard Bass University of Connecticiut  (PG) These are notes for a course that is becoming increasingly common due to the new demands from financial engineering. It's pitched at the right level-only a good post-calculus probability course is needed as prerequisite. It's rigorous,concise and well written-although it could use more examples to help flesh out this fascinating and conceptually demanding subject. A good introduction to the subject for undergraduates.

Probability and Stochastic calculus, MSFI Tristan Tomala, HEC Paris (PG)  A very minimalist primer of the basic elements of probability for finance students. Not good for much else.

 

Stochastic Processes (Applied) Bruce Reed McGill University 2008 (PG) This is an applied course in stochastic processes that emphasizes examples and applications while avoiding measure theory. While clearly not a course for serious mathematics students, it's careful and gives many good examples, particularly of the use of generating functions in solving stochastic differential equations and an introduction to percolation. A good undergraduate course or course for non-mathematics graduate students.

 

Stochastic Processes  Russell Lyons Indiana University 2012  (PG-13) Although technically for undergraduates, these notes are more comprehensive and mathematically oriented then one would expect from such a course.Although they don't explicitly require measure theory, they do require a good command of rigorous calculus/ elementary real analysis as the convergence and limit theorems of random processes are done very carefully. Many examples are given and the general concepts of stochastic processes are discussed very clearly and in detail by building on a first course in probability to supply basic concepts. For example, renewal theory is presented as a generalization of the Poisson process. Still, it's a very challenging set of notes and probably better suited for first year graduate students then undergraduates except at very strong undergraduate programs. Recommended for the right audience.

Stochastic Processes Queens University At Kingston (PG)These are notes for an applied course in stochastics with minimal prerequisites . They are concise but are clear and give many examples and computations. Again, good for an undergraduate course. 

 

Stochastic Processes I Donald Estep Colorado State University Course Materials (PG-13) This is another of Estep's excellent handwritten course notes. Pitched as an applied course for graduate students in statistics, it requires about the same prequisites as Lyon's course above ( i.e. undergraduate probability,linear algebra and a good course in elementary real analysis or honors calculus). However, it is much more self contained and Estep reviews most of the needed concepts as he goes. Some basic probability results-such as standard theorums on conditional probabilty- are stated and used without proof. Conditional probability,in fact, is covered in detail before any actual Markov processes are described. This is a very good idea in a course like this since the entire concept of a stochastic process really hinges on it and conditional probability is not always covered in detail in elementary probability. Markov chains are then defined as sequences of conditionally dependent random variables whose conditions are "local". These notes are just as clear and insightful as they develop all the basics-random walks, discrete Markov chains, transition matrices and absorbtion times, continuious Markov chains and more. There are many examples and best of all, Estep's handwriting is very clear and legible and this is critical when you're working entirely off scans of handwritten notes. One of the best free introductions available.Very highly recommended.

 

PROBABILITY AND STATISTICS STEFAN WANER BASED ON G.G ROUSSAS A COURSE IN MATHEMATICAL STATISTICS (PG) A very nice supplementary set of notes for a basic post-calculus course in probability and statistics. Relatively brief and isn't very detailed, but all the basic definitions and theorums are covered with some nice examples. But be warned it's not really intended as a course text and it doesn't have enough detail for that. A useful study aid when combined with a more detailed presentation.

 

Essentials of Stochastic Processes Rick Durrett Duke University (PG-13) The finished second edition of the book has since appeared published by Springer-Verlag in 2013, but Durrett has very kindly kept the "nearly complete" draft available freely at his website. Similar in content to Estep's notes, but with much more detail and many more examples and graphs. There's also much more emphasis on functions of several variables with random vectors and the transition matrix from jump.In addition to the usual topics, it has chapters on martingales and applications to financial mathematics. It's also surprisingly error free for a draft of a book.  This is a masterly presentation of the basics by one of the top researchers in probability and he's done a huge service continuing to make this flawed but still excellent draft available. Very highly recommended. Probability and Statistics Anwar Hossain and Oleg Makhnin New Mexico Tech University 2013 (PG)Fairly standard, albeit detailed set of lecture notes for the post-calculus probability and statistics sequence. Lots of solved problems and examples and all the basics discussed with careful proofs given of the main results, but the emphasis is clearly on problem solving techniques. The main highlight of these notes is that it gives particularly complete presentations of the basic probability distributions, including several that usually aren't discussed in a first course, like the negative binomial distribution and the gamma distribution. Also, quite a few good exercises.A good set of notes and useful for a study aid in a first course. Recommended.

Probability and Random Variables Scott Sheffield MIT Spring 2011 (PG)Lecture notes for a standard undergraduate course in probability. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem." Very detailed with many exercises and insights, also very careful and rigorous for a post calculus course. Recommended.

Probability Theory Course Notes Curtis McMullen Harvard University May 4, 2011 (PG-13) These are McMullen's course materials for an undergraduate course at Harvard University pitched at much stronger students then the average university fills up it's course with. The notes are based on the classic 2 volume text by Feller (!), which indicates a seriously theoretical treatment without measure theory.The notes are beautifully written, comprehensive and detailed with an emphasis on proving the major results of probability functions and random variables up to the Central Limit Theorum. But the notes aren't devoid of examples-indeed, McMullen discusses some inventive applications, including the distribution of bombs that fell in London during the Second World War, Benford's law of the approximation of smooth distributions over a very large range by the normal distribution and the optimal stopping problem. An incredibly rich and wonderful resource-like all of McMullen's course materials-and is highly recommended to all serious students of probability.

Undergraduate  Probability  Davar Khoshnevisan University of Utah, Spring 06(PG)Clear, but rather scattershot and disorganized lecture notes to accompany a course based on Ross. Wierdly, the lecturer didn't bother to create pdfs of the scanned scribblings, which tells you something. They clearly aren't intended to act as the main source for the course, but to give a few extra examples and insights. Worth a look,but not my favorite.

Statistics William G.Faris University of Arizona December 1, 2003  (PG)Very comprehensive and well written set of lecture notes on qualitative statistics for a follow up course to a post-calculus probability course. Covers the standard topics such as estimation, hypothesis testing, order statistics such as the median and goodness of fit models, variance and likehood function estimation and more. Rigorous with many examples and good problems, as well as discussions illuminating the underlying ideas of statistics.Another excellent resource I wish I knew about when learning statistics. Highly recommended .

Mathematical Statistics Jan Vrbik  Brock University  (PG)These are lecture notes for a one semester undergraduate course in statistics pitched at roughly the same level as Faris' notes. Not as comprehensive, but very clear, careful and with many examples.The author manages to pack a lot into 84 pages while still giving sufficient detail for self study. It also includes several topics of probability that sometimes aren't covered in the usual course, but are very important in mathematical statistics, such as the Cauchy distribution and transformation of random variables. It also includes a final chapter on nonparametric statistics, which there sometimes isn't time for in a one semester course. Students that like notes that give all necessary details but no more will like these a great deal. Again, a set of notes I wish I'd had when I was learning the subject. Highly recommended.

Probability and Statistics Dmitry Panchenko MIT Spring 2005 lectures (PG) Another version of the undergraduate course in probability and statistics at MIT. Terse and incomplete, no proofs, but contains many insights and instructive solved problems. Really intended as a supplement rather then the main course text. In that context, they are very helpful indeed. Couple with a good text or comprehensive lecture notes, they are very good indeed. Recommended.

Probability 2012/2013 Robert Johnson University of Cambridge (PG)  Excellent if somewhat terse course notes and exercises for the undergraduate post-calculus course at The University of Cambridge.Like most mathematics notes at Cambridge, they focus more on theory and far less on applications then American notes. There are also many challenging and thoughtful exercises sheets. Logic, set theory, sample spaces, probability functions and the axioms of probability, distributions and random variables. Unfortunately, by the author's own admission, they are only a summary of what's covered in class and several sessions notes are missing. It would be a real find if one of his former students posted a full transcript of his or her own notes from the actual course. As they stand, though, they're good as a supplement and nothing more. Recommended but limited.

Probability I Robert Johnson Queen Mary College The University of London 2010 (PG) A virtually identical course given in 2010 at Queen Mary by the same author as the immediately preceeding notes. Same comments apply.

Probability II 2009/2010 I. Goldsheid Queen Mary College Univeristy of London  (PG)Course notes and materials for the follow up course to the one taught by Johnson above. They are very similar in nature but somewhat more detailed, covering more advanced material in probability including conditional probability,Markov chains and branching processes, and multivariable probability functions and joint distributions. They also come with many good exercise sheets. Very well written and careful, a very good study source for students in a good undergraduate probability course. I still think they don't have enough detail to be used as a main course text, though. Recommended.

Probability Theory Semester 2 Boris Tsirelson, Tel Aviv University  2005/2006 (PG-13)  An interesting and different set of notes from Isreal. They're notes for a second course (sadly, the first semester notes are no longer available at the course website). They're pitched at a higher level than most undergraduate notes on the subject, but develop multivariable density and distribution function theory without explicit measure theory. Instead, Tsirelson uses the old theory via Jordan measure. This allows him to develop virtually the entire structure of modern probability rigorously by ensuring all boundary sets have measure zero. It's a good, creative approach,but I'm not sure it's easier then just developing the measure theory. Also, it's pretty lacking in exercises and examples. Recommended for the theoretically minded student who's looking for something different,but someone looking for just something to help them learn probably should look elsewhere.

Martingale Ideas in Elementary Probability Independent University of Moscow Spring 1996 William Faris University of Arizona Fulbright Lecturer 1995{1996(PG-13?) An interesting, very concrete,  example  driven but rigorous  introduction to stochastic processes. While it does use the basic definitions of measure theory to rigorous construct probability spaces, most of the notes really focus on examples of martingales and Markov chains and how to compute with them. Very well written with many examples and exercises. However, the level of difficulty here is a bit tough to decide as the notes vary quite a bit depending on subject. A student with a good honors calculus course should have no trouble following them,but they're clearly more demanding then other undergraduate introductions to the subject. Perhaps to play it safe, I should have put it in the graduate section. Oh well, maybe in future updates. Recommended for strong students.

Probability Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology FALL 2000 (PG)This is an early draft of the very popular book on introductory probability aimed at engineering students. I'm told it has many errors in it, but from what I can see, the errors are mostly obvious ones and are in the exercises. They can be worked out by students. It clearly emphasizes techniques of solution and computation,with many, many examples. But all the major theorems at this level are proven carefully and all concepts are stated rigorously, using no more background then calculus.  If you can't afford the finished book, this is a very good substitute and I'm glad the authors have continued to make this early draft available.Recommended. 

Probability Models Chris Morgan Purdue University Statistics  (PG)An excellent, very detailed and visual set of notes on applied probability and statistics that contains a wealth of material for a basic undergraduate probability course, beginning with set theory and going through basic probability and statistics. The notes have tons of exercises. The notes are in Power Point format, which is both a blessing and a curse. The blessing is that many beautiful and informative graphics can be added which greatly enhance the presentation. The curse is that they're harder to open and download. Unfortunately, the notes have one gigantic drawback for mathematics majors-virtually no proofs. If the author decides to rewrite these notes one day with theoretical material, this will be one of the very best sources on the web for first courses in probability and statistics. As it stands, it really can't be used by math majors as anything but a supplement. That being said, it's one of the best supplements on the web and will make a fabulous study tool for practice and drill. Combined with a more rigorous presentation-such as one of the Oxford lecture sets-it would form the basis for an outstanding free textbook in first year probability and statistics. But sadly, since it lacks proofs, it'll be of limited use by itself. Still, recommended. 

Elementary Probability William G. Faris University of Arizona February 22, 2002 (PG) The prerequisite notes to Faris' statistics notes above. They are just as well written, detailed and careful as the sequel and contain several original quirks, such as an unusually detailed undergraduate level discussion of the weak Central Limit Theorum and how it motivates the techniques of centering and standardizing distribution functions. Many examples. Unfortunately, a huge drawback is that unlike the sequel, there are no exercises in this set of notes. But these are easily obtained elsewhere.(Indeed, the union of these notes and Moran above would form an amazing free online text for introductory probability!)  Again, a set of notes I wish I'd had as an undergraduate and a great study aid for a first course. Highly recommended.

 

A Short Introduction to Probability Dirk P. Kroese  University of Queensland   (PG) This is the set of notes I was hoping Moran's would have been if he included proofs of all the theorems. Deep and comprehensive, with an huge number of examples illustrated in color. Kroese gives many insightful sidebars to clarify concepts-for example, he explains the expectation several different ways, including as a kind of "expected profit" of a long series of bets. Despite the "short" title description, an enormous amount in packed into these lectures and a student who works through them will come away with a terrific understanding of random phenomena. Unfortunately, there are very few exercises and they are packed at the rear. As I've said elsewhere, though, these are easily supplied from other sources. An outstanding set of undergraduate lectures with many good examples. Very highly recommended.

Probability I (Spring 2012) James Pascaleff Universty of Texas (PG)Very nice, detailed set of handwritten notes for a standard undergraduate post-calculus probability course. Clear and legible, which is always a plus. Coverage is very standard- Pascaleff sticks very close to the usual layout of a basic course. The notes give a particularly detailed examinations of basic combinatorics and it's relation to discrete probability as well as the elements of multivariable probability densities and distributions. Not going to replace Ross or Feller, but very solid and definitely worth a look. Recommended.

Probability Part A Christina Goldschmidt University of Oxford  Michaelmas Term 2015  (PG-13) The first semester of the 2 term undergraduate probability sequence at the University of Oxford-which means it's uncompromisingly rigorous and careful while not neglecting conceptual intuition or applications.Probability measures are defined and used to carefully define probability spaces, but abstract measure theory is not covered.Both discrete and continuous random variables, densities and distributions are covered in great detail, including the gamma and Bernoulli distributions. More nonstandard material covered includes the solution of first and second order linear difference equations,random walks on fi nite state spaces, probability generating functions with applications  in calculating expectations, the Chebyshev's inequality and a careful proof of the Weak Law of Large Numbers.Well written with many examples, this is a typical Oxford set of notes: Detailed, challenging and extremely informative. It will also serve as a perfect complement to a more applied set of notes like Moran. Highly recommended.

ProbabilityPart A James Martin Oxford University Hilary Term March 2013   (PG-13) A second course in probability at the undergraduate level at Oxford, the follow up course to Goldschmitt's above.  Martin covers first convergence of random variables, generating functions and a deeper study of multivariable probability densities and distributions. The rest of the notes' chapters give an in-depth undergraduate level presentation of stochastic processes, including Markov chains, stationary processes, transition matrix computations and Poisson processes. Martin's notes are as careful and well written as Goldschmidt's and will be of great help to serious undergraduates trying to learn stochastic processes. Combined, Goldschmidt and Martin will provide the basis for an outstanding year long course in probability for strong mathematics students.  Highly recommended.

Statistics Neil Laws Oxford University Hilary Term 2014

Statistics Neil Laws Oxford University Hilary Term 2012 Slides.pdf (PG) This is the follow up course in statistics with the Part A course in probability at Oxford as the prerequisite. There are 2 sets of lecture notes, both equally important and different-the first is the "standard" notes drafted by Laws from the previous iterations of the course and the resulting notes of other instructors. The second "slides" notes are the notes he himself wrote for the specific class he taught. Statistics and parameters, estimates and estimator functions, Q-Q plots,confidence intervals and goodness of fit models. The notes aren't very comprehensive, but what they do cover, they do an outstanding job of presenting. They are very careful and they do something many books and notes on the subject fail to do: motivate the subject from a mathematical perspective. Laws makes the excellent point at the beginning that in many ways, statistics is the opposite of probability: Instead of having a known parameter for which the values form the range of a random variable as in probability,in statistics, we use data to establish bounds on the values of an unknown parameter, which in turn allows us to determine information about the probability distribution of the values of the data.  An excellent introduction to the subject and Law's slide notes provide many more illustrative examples. Both are highly recommended.

Probability and Random Variables Middle East Technical University 2012 (PG)Average set of notes for a standard introduction to probability. Lots of  examples but also lots of holes in the notes where presumably students where to fill in proofs and missing calculations. Useful for practice mainly, but not much else.

Probability and Statistics for Computer Science Lecture notes for Statistics Hal Stern University of California, Irvine  (PG)   Very shallow, almost bullet point notes on basic probability theory and statistics clearly not intended to supply a comprehensive introduction to the subject. Might be ok for review, but not much else.

PROBABILITY Nathaniel Derby Summer 2009 University of Washington  (PG) Very comprehensive and rigorous set of lecture notes at the undergraduate post-calculus level. Very good pictures and many examples and solved problems, making them very suitable for self study. My one quibble is that basic combinatorics isn't covered in the text proper, but in an appendix. This has the advantage of getting right to discrete probability, but unless the students had a background in either discrete math or combinatorics,they may get left behind. Still, one of myfavorites and wish I'd had them when I was learning probability. Very highly recommended. .

PROBABILITY Fall 2007  Joshua M. Tebbs University of South Carolina  (PG)Another rigorous, well-written and comprehensive set of lecture notes at the undergraduate post-calculus level. Very similar in coverage and style to Derby's,but not quite as thorough or user friendly. Unusually strong in presentations of conditional probability, basic combinatorics and the various probability distribution functions.An excellent resource, recommended.

Probability G. A. Young Imperial College September 2011  (PG)Typical British "Cambridge" style lecture notes for an undergraduate course: rigorous, very concise and blunt while being short on examples. Still, well written and pitched at a higher level then usual course notes in the US. Recommended for strong students.

 Probability and Statistics:The Digital Textbook Marco Taboga  (PG) This is a weird text that's wortjh  checking out-it's a free online textbook written and organized by Marco Taboga, who's an applied mathematician and finance analyst for the Bank of Italy. It covers essentially all the material in an undergraduate course on basic probability and mathematical statistics as can be covered without measure theory. The book emphasizes problem solving techniques and contains more then the usual coverage of probability distributions and statistical methods, including the gamma and Wishnart distributions. A very nice, readable text broken into bite size pieces and completely free. So even if you're not using as the main text, why wouldn't you keep a copy handy for you or your students? Highly recommended.

THE EXPECTATION PRIMER EXPECTATION COVARIANCE  Probability and statistics (Undergraduate) Maurice Joseph Dupré Tulane University SPRING 2010  (PG)This is a very original undergraduate presentation of probability and statistics by Dupre built around the concept of conditional expectation and deriving the basic properties of the probability measures and thier distribution functions from it. The central idea is that probability and statistics are the study of the various measures of "guessing" of results (like guessing what faces a specific thrown die land on most of the time), This requires a precise formulation of the idea of frequency or central tendency and the mathematically precise definition is expectation depending on certain random variables. This striking and very creative approach leads to a very interesting presentation of the standard probability/statistics course in a dramatically nonstandard way. Dupre makes it clear in the syllabus these lectures aren't intended to function independently of a standard course text, but are intended to give an alternate motivation for the ideas. I completely agree- I'd certainly recommend it as collateral reading for students and instructors alike.

Introduction to Probability and Statistics Dmitry Panchenko Spring 2005  (PG) Yet another of MIT's open course ware versions of the basic post-calculus probability course. Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.This version is a bit more complete then the other versions with nice exercises and examples. Good but nothing that'll change your life.

Probability and Statistics in Engineering MIT OpenCourseWare (PG) Exactly what the title and the course description says it is. "This class covers quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, random processes, statistics, and decision analysis are covered, along with random variables and vectors, uncertainty propagation, conditional distributions, and second-moment analysis. System reliability is introduced. Other topics covered include Bayesian analysis and risk-based decision, estimation of distribution parameters, hypothesis testing, simple and multiple linear regressions, and Poisson and Markov processes. There is an emphasis placed on real-world applications to engineering problems."I think such courses, no matter how well written or comprehensive, are almost always best used as supplements to more theoretical treatments-supplying practical examples to balance theory. These don't look like an exception. Good for undergraduates in the sciences.

 Probability R. Vershynin University of Michigan Winter 2012 (PG)A standard  undergraduate course based on Ross with very good supplementary notes featuring some not so standard topics, like the matching problem. Not comprehensive, but very clear and lucid and will help in any introductory course. Recommended.

Probability Wai Kong (John) Yuen Brock University Fall 2013 (PG)A very lucid and occasionally amusing (!) set of notes for a standard post-calculus probability course. Not only are they careful and surprisingly complete for a comparatively brief set of notes,  but entertaining at times-the author lists as one of the goals of the course to ensure the students "never walk into a casino again ever!" Definitely worth a look. Recommended.

Probablility (Undergraduate level) Richard Bass University of Connecticiut 2013: (PG) One of Bass' usual sets of lecture notes: concise, lucid and with all the essential points.He also has some nice examples and a good selection of topics, such as a whole chapter on common distributions..But unlike his graduate notes, there's only enough material here for a one semester course. Make sure you get the most recent editions (2013 and later) -the earlier versions are too terse to be used as anything but a supplement.  Indeed, that may be how Bass originally intended them to be used before expanding them. Recommended.

Probability And Statistics With Applications Rick Dilling University of Tennesee (G)An scant  outline containing mostly applications of statistical analysis to biology.Completely lacks details that presumably were filled in class. Good as a source for real life applications,but nothing else. . 

A Probability Course for the Actuaries A Preparation for Exam P/1 Marcel B. Finan Arkansas Tech University Preliminary Draft 2012  This somewhat nonstandard course in undergraduate probability is another fine online textbook by Finan. It also fills a lacuna of online sources available to students preparing for the all-important and incredibly intense series of examinations.

Notes on Probability Peter J. Cameron Queens College of The University of London  (PG)This is another of Cameron's wonderful lecture note sets, very much in the Cambridge/ Oxford UK university tradition. It covers all the basics of an undergraduate probability course with great care, detail and rigor, with a level of difficulty perfect for such a course. At the same time, Cameron provides many insights and examples from genetics, gambling, electrical circuit theory and more. It's also beautifully written, with many quotations from mathematicians and scientists on the science of probability, which flat out makes the notes a joy to read. Each chapter ends with a number of problems with complete solutions. The one flaw in the notes,sadly, is not enough exercises. But again, this is easily solved with supplementary problems-Cameron himself gives a good list of references, including Grinstead and Snell's excellent online textbook. A terrific source that both educates and entertains. Very highly recommended.

Lectures on Statistics Robert B. Ash Professor Emeritus, Mathematics University of Illinois

Probability and statistics Manjunath Krishnapur Indian Institute of Science Fall 2013 (PG) Very concise notes for an undergraduate course with broad but  very standard coverage.Nicely written with good problems, but nothing here you can't find in more substantial sources online. You may like them, but I wasn't impressed.

Elementary Probability Course Notes Tom Salisbury York University Winter 2010 (PG)Very nice, "bullet point" style notes for an undergraduate course. Unfortunately, they're really intended to supplement a text rather then function on their own, so most results are stated without proof and they aren't as complete as many of the other sources listed here,That being said, they state definitions very carefully and give many insightful examples and descriptions. A very helpful study aid in addition to a text. Recommended.

Introduction to Probability Charles M. Grinstead Swarthmore College J. Laurie Snell Dartmouth College    (PG)This is the online version of the wonderful and popular textbook, which Snell and Grinstead have blessed us with by keeping it available for free download on the Dartmouth website via The CHANCE project. It is justifiably popular, written by 2 experts in the field for a very broad audience ranging from honors high schoolers to graduate students in other fields who are weak in probability theory. This remarkable book is, in many ways, an updated and much gentler undergraduate version of the first volume of William Feller's magnum opus on the subject. Indeed, Feller's book is credited by both authors in the preface as their inspiration. The book is one of the smoothest and most lucidly written textbooks I've ever seen on any subject and I wish it was available when I was learning the subject. It's not what one would call rigorous, but it is quite careful. The book has 2 real strengths that make it both very original and immensely useful to a very broad audience: Firstly, it has a ridiculously large number and diversity of detailed examples, exercises and solved problems, an absolute must for any student attempting to learn a field as vast and important as probability. The examples range from something as simple as a fair coin flip to something as complicated as a computer generated model for various random walk processes.Computer programs are fluently used, as is careful calculus problem  solving, as tools in the examples and exercises So there's something for everyone in this book. Secondly and more importantly is the rather unorthodox structure of the book-discrete and continuous probability distributions and functions are covered in independent chapters. This makes the book enormously flexible-it can be used for various kinds of undergraduate courses, from a one semester discrete probability course to a full year course in calculus-based probability theory. My quibble with the book is that in it's passion to make the subject as clear and comprehensible as possible, a number of important topics are omitted, such as the role of the gamma function. But these are just quibbles. Overall, this is one of the very best currently available textbooks for undergraduates.The fact it's available freely not only makes it a must have for both students and teachers of probability, it demonstrates the future that's possible if we embrace this format and encourage many more such books to be written and made available in this manner.  Very highly recommended for beginning students in probability.

Probability and Statistics M. H. Faber  ETH  Zurich  2007  (PG)This is an interesting set of notes for an undergraduate probability and statistics course aimed at engineering students.As would be expected from notes aimed at  engineering students, they're very applied and rigor isn't really a concern. That being said-the notes give fascinating insights into the practice and perception of probability and statistics in the "real world"-the distinction between classical frequency interpretation of probability and the Baylesian formulation (which frankly, has always mystified me), the use of estimation models and numerical random approximation in evaluating solutions and risk in building projects and much more. These notes would make wonderful collateral reading alongside a more rigorous treatment for anyone who either studies or uses  probability and statistics. Recommended.

Probability Theory Course Notes and Class Supplements Curtis McMullin Harvard University 2011 (PG-13) A beautiful but very intensive set of lecture notes by the famed researcher and teacher at Harvard, clearly  designed for superior students at the undergraduate level. Contents  I The Sample Space  II Elements of Combinatorial Analysis III Random Walks IV Combinations of Events V Conditional Probability VI The Binomial and Poisson Distributions VII Normal  Approximation VIII Unlimited Sequences of Bernoulli Trials IX Random Variables and Expectation  X Law of Large Numbers XI Integral–Valued Variables. Generating Functions XIV Random Walk and Ruin Problems I The Exponential and the Uniform Density II Special Densities. Randomization . .They cover all the basics of probability theory from basic sample space concepts through discrete and continuous random variables, probability density and distribution functions, expectation, generating functions, random walks and much more-all packed into a very dense 98 pages. They are intended to supplement a course based on the first edition of William Feller's classic, which gives a good idea of the kind of class we're talking about. Still, the notes are extremely readable if care is taken reading them and the author adds many unusual and wonderful applications, including the birthday problem, roullette wheel prediction, The Two Envelopes problem and much more. While it's clear this is a terrific source for a probability course, it's also clear it's for serious students only. But for such students, it will prove extremely valuable as a supplement to a standard text. Highly recommended.

Probability IA Douglas Kennedy Trinity College Cambridge 2010:  Another excellent set of notes and exercises for a semester long undergraduate probability course at Cambridge. Careful, readable and with many examples and good exercises. Also includes some unusual inclusions such as a detailed discussion and proof of the Sterling formula and the Simpson paradox. A terrific source. Highly recommended.

CHOICE ,CHANCE, AND INFERENCE An Introduction to Combinatorics, Probability and Statistics Carl Wagner University of Tennessee    One of the most original and interesting sets of lecture notes I've seen-it's clearly a book in progress.These notes give a unified presentation of basic combinatorics and both discrete and continuous probability theory at the post calculus level-and giving basic statistical analysis methods as applications of the preceding theory. God, I wish I'd known about these notes when I was taking these courses.  The result would have been so much better.