# 5 Free Textbooks on Probability Theory and Its Applications

Probability Theory is a branch of mathematics that provides a quantitative measure of the likelihood of the occurrence of an event. When dealing with uncertain situations, you can weigh the potential outcomes using probability theory to make informed decisions. Numerous books on Probability Theory are available online, but finding quality resources can be time-consuming.

To make the search easier for you, I conducted extensive research to find the best free textbooks on probability theory and its applications. Initially, I created a list based on average ratings from Amazon. From that list, I further shortlisted books based on community preferences and my own inclinations. I manually collected feedback from various sources to ensure that the selected books were not only of high quality but also highly relevant to the needs of both beginners and advanced learners.

So, without further delay, let’s get to this final list.

## 1. Probability Theory: The Logic of Science by E. T. Jaynes

The book Probability Theory: The Logic of Science stands out at the top of my list. It goes beyond the conventional mathematics of probability theory by providing deeper and more intuitive explanations of the concepts rather than presenting them as axioms only. The book also focuses on the practical applications. A major strength distinguishing this book from others is its inclusion of historical perspectives, which enhances the understanding of the subject. Additionally, the book contains numerous exercises for practice purposes. If you are someone who is not only interested in the “how” but also the “why” of any concept, then this book is a perfect option for you.

### Book Content

Part I: Principles and elementary applications

1. Deductive and Plausible Reasoning, Quantitative Rules, Sampling Theory, Hypothesis Testing
2. Queer uses for Probability Theory, Parameter Estimation, Central, Gaussian, or Normal Distribution
3. Sufficiency, Ancillarity, Probability and Frequency, and Physics of Random Experiments

1. Discrete Prior Probabilities, Ignorance Priors and Transformation Groups, Decision Theory
2. Paradoxes of Probability Theory, Orthodox Methods: Historical Background, Orthodox Statistics
3. Physical Measurements, Model Comparison, Outliers and Robustness, Communication Theory

## 2. Introduction to Probability by Joseph K. Blitzstein & Jessica Hwang

Introduction to Probability is an excellent introductory book that is famous for its story-like writing style with real-world examples to clarify concepts. It is more popular among students and self-learners because you can also find the lectures for this course. Additionally, the book includes 600 practice problems of varying difficulty, many of which go beyond simple plug-and-chug exercises. The R code at the end of each chapter allows readers to run simulations and check their understanding through hands-on experimentation. The author not only explains a solution but also comments on how they knew to take the approach they did. Another thing that I personally liked about this book is that it also dispels the common myths and misunderstandings that are marked by the (☣) biohazard symbol.

### Book Content

1. Probability and Counting, Conditional probability, Random Variables, and their Distributions
2. Expectation, Continuous Random Variables
3. Moments and Joint Distributions
4. Transformations, Conditional Expectation, Inequalities, and Limit Theorems
5. Markov Chains, Markov Chain Monte Carlo, Poisson processes

Check out the accompanying course material here: Statistics 110: Probability Youtube Lectures

## 3. Probability and Statistics for Data Science by Carlos Fernandez-Granda

The author of the book Probability and Statistics for Data Science is an esteemed Associate Professor of Mathematics and Data Science at NYU. This book is actually a collection of the course notes for the probability and statistics course taught as part of NYU’s master’s in data science program. It is an excellent resource for anyone looking to refresh their understanding of fundamental concepts in probability and statistics, especially from a data science perspective. Despite its high quality, these notes have surprisingly received little attention, so I wanted to share them. You can also access the lecture videos from the author’s YouTube channel, for which the link has already been shared above.

For the full learning experience related to this book, find the accompanying YouTube lectures.

### Course Content

1. Basic Probability Theory, Expectation, Random Variables, and Processes
2. Markov Chains, Descriptive Statistics, Frequentist Statistics, and Bayesian Statistics
3. Hypothesis Testing, Set Theory, Linear Regression, and Linear Algebra

## 4. Probability: Theory and Examples by Rick Durrett

The book Probability: Theory and Examples provides a practical introduction to probability theory. The author’s philosophy is that the best way to learn probability is through examples, and this book includes 200 examples and 450 problems to illustrate key concepts in action. Additionally, the inclusion of “tricks” that may not be easily found elsewhere is a valuable bonus. While it may not be suitable for self-study as a first introduction to probability, but is an excellent resource for someone looking to apply their understanding of this subject to real-world problems.

### Course Content

1. Measure Theory, Laws of Large Numbers, and  Central Limit Theorems
2. Martingales, Markov Chains,  Ergodic Theorems, and Brownian Motion
3. Applications to Random Walk, Multidimensional Brownian Motion, and Measure Theory Details

## 5. Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik

The beginner-friendly book Introduction to Probability, Statistics, and Random Processes covers both probability theory and statistical methods, making it an excellent resource for students and practitioners alike. Many readers consider it a perfect mix of theory, exposition, and examples. The digital version of this book is also available, along with supplementary materials like videos. This effort to make the content accessible to more people is truly commendable.

The way the content is structured and presented makes even complex topics much easier to grasp. However, some advanced topics are covered briefly, and you may need to consult additional resources to fully understand those sections. Overall, this book is a great pick for its clear explanations and practical insights. It’s definitely worth checking out.

### Course Content

1. Basic Concepts: Random Experiments, Probability Axioms, Conditional Probability, and Counting Methods
2. Single and Multiple Random Variables (Discrete, Continuous, and Mixed), Moment-generating Functions, Characteristic Functions, Random Vectors, and Inequalities
3. Limit theorems and Convergence, Bayesian and Classical Statistics
4. Random Processes, Processing of Random Signals, Poisson Processes, Discrete-time and Continuous-time Markov Chains, and Brownian Motion
5. Simulation using MATLAB and R (online chapters)

That’s it for today’s article. Suppose you are short on time and a beginner. In that case, I recommend starting with Introduction to Probability by Joseph K. Blitzstein & Jessica Hwang and Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. Both of these books also have video content available for better understanding. For more advanced learners, if I had to pick a single book, it would be Probability Theory: The Logic of Science by E. T. Jaynes. This book goes beyond conventional mathematics, offering deeper and more intuitive explanations.