Lectures on Probability Theory and Mathematical Statistics

by Marco Taboga

This book is a collection of lectures on probability theory and mathematical statistics. It provides an accessible introduction to topics that are not usually found in elementary textbooks. It collects results and proofs, especially on probability distributions, that are hard to find in standard references and are scattered here and there in more specialistic books.
The main topics covered by the book are as follows.
PART 1 - MATHEMATICAL TOOLS: set theory, permutations, combinations, partitions, sequences and limits, review of differentiation and integration rules, the Gamma and Beta functions.
PART 2 - FUNDAMENTALS OF PROBABILITY: events, probability, independence, conditional probability, Bayes' rule, random variables and random vectors, expected value, variance, covariance, correlation, covariance matrix, conditional distributions and conditional expectation, independent variables, indicator functions.
PART 3 - ADDITIONAL TOPICS IN PROBABILITY THEORY: probabilistic inequalities, construction of probability distributions, transformations of probability distributions, moments and cross-moments, moment generating functions, characteristic functions.
PART 4 - PROBABILITY DISTRIBUTIONS: Bernoulli, binomial, Poisson, uniform, exponential, normal, Chi-square, Gamma, Student's t, F, multinomial, multivariate normal, multivariate Student's t, Wishart.
PART 5 - MORE DETAILS ABOUT THE NORMAL DISTRIBUTION: linear combinations, quadratic forms, partitions.
PART 6 - ASYMPTOTIC THEORY: sequences of random vectors and random variables, pointwise convergence, almost sure convergence, convergence in probability, mean-square convergence, convergence in distribution, relations between modes of convergence, Laws of Large Numbers, Central Limit Theorems, Continuous Mapping Theorem, Slutski's Theorem.
PART 7 - FUNDAMENTALS OF STATISTICS: statistical inference, point estimation, set estimation, hypothesis testing, statistical inferences about the mean, statistical inferences about the variance.

Genres: Mathematics, Nonfiction

656 pages, Paperback

First published December 8, 2012

Reviews

[Rebecca · Rating 5 out of 5]

I'm not going to lie- I didn't actually read this. I tried, but I don't know enough about math to appreciate it. It was a good reads giveaway, and I thought it was more like just lectures.

That said, I gave it to my sister who has a degree in mathematics, and she started reading it right there at the dinner table and thought it was a well written and informative text.

So she gave it to her fiance, who has a physics degree and he "loved it" and appreciated the gift. I have asked her to write a review when she gets the chance, but in the meantime I wanted to let everyone know that 2 impartial sources thought it was a great book!

[Bryn Collier · Rating 5 out of 5]

This is an extremely good "intermediate" prob stat textbook. It does not cover linear regression, and most of the book focuses on probability theory. Each chapter is short and self-contained and references previous sections whenever mentioning a vaguely unfamiliar concept. Mathematically, the book runs a middle course. It is formal enough to avoid creating gaps while informal enough to be digestible. In this sense, it is "pragmatic" - it expects some mathematical commitment from the reader but does not expect them to breathe in symbols only. There are some advanced topics that you wouldn't find in a standard undergraduate textbook (Lebesgue integrals, basic measure theory) but it's more to acquaint the reader with the concepts and how they apply to probability if they are interested. Tl;dr, if you don't like walls of text and have a reasonably strong mathematical background (calculus, linear algebra, basic proofs), this is the right textbook for you.

[Melissa Herston · Rating 4 out of 5]

In compliance with FTC guidelines; I received the book for free through Goodreads First Reads.

Lectures on Probability Theory and Mathematical Statistics is an excellent text, because it is clearly written, easily readable, covers a lot of ground, and explains things intuitively.

After going through this book and looking at some other comparable titles, I have to come to the conclusion that this is an excellent text on the topic of Probability Theory and Mathematical Statistics. At 600 plus pages the book is quite long, but the sheer amount of material covered is simply astounding.

This book is well-organized and a good reference text. The book does not assume that the reader is familiar with the topic, but rather starts with the simplest solutions and builds from there. I felt the book is self-contained enough for readers who have not heard of the topic before to get a good idea of it.

There are a variety of well-designed problems that provide plenty of practice along with some that expand upon the original concepts, and the average difficulty generally seems about right for the target audience. The author goes to great lengths to make every concept clear to the reader.

This is a book for those who are interested in probability theory and want to gain a broader understanding of the subject matter. This book is well-organized and is one of the clearest and most readable books on the topic. This book is very full and is a very good resource on the topic.