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Special Features Table of Content · Introduction: The Nature of Probability Theory · The Sample Space · Elements of Combinatorial Analysis · Fluctuations in.

**Table of contents**

- An Introduction to Probability & Statistics
- Experiment, Sample Space and Event
- An introduction to probability theory and its applications.
- Introduction to Probability

Probability Spaces.

Random Variables 5. The Extension Theorem 6. Product Spaces.

Sequences of Independent Variables 7. Null Sets. Distributions and Expectations 2. Preliminaries 3. Densities 4. Convolutions 5. Symmetrization 6. Integration by Parts. Existence of Moments 7. Further Inequalities. Convex Functions 9. Simple Conditional Distributions. Mixtures Conditional Distributions Conditional Expectations Stable Distributions in R 1 2. Examples 3. Infinitely Divisible Distributions in R 1 4. Processes with Independent Increments 5. Ruin Problems in Compound Poisson Processes 6. Renewal Processes 7. Examples and Problems 8. Random Walks 9. The Queuing Process Persistent and Transient Random Walks General Markov Chains Martingales Applications in Analysis 1.

Main Lemma and Notations 2. Bernstein Polynomials. Absolutely Monotone Functions 3. Moment Problems 4. Application to Exchangeable Variables 5. Generalized Taylor Formula and Semi-Groups 6. Inversion Formulas for Laplace Transforms 7. Strong Laws 9. Generalization to Martingales Convergence of Measures 2. Special Properties 3. Distributions as Operators 4. The Central Limit Theorem 5. Infinite Convolutions 6. Selection Theorems 7. Ergodic Theorems for Markov Chains 8.

Regular Variation 9. Asymptotic Properties of Regularly Varying Functions Orientation 2. Convolution Semi-Groups 3.

Preparatory Lemmas 4. Finite Variances 5. The Main Theorems 6. Example: Stable Semi-Groups 7. Triangular Arrays with Identical Distributions 8. Domains of Attraction 9. Variable Distributions. The Three-Series Theorem The Pseudo-Poisson Type 2. A Variant: Linear Increments 3. Jump Processes 4. Diffusion Processes in R 1 5. The Forward Equation. Boundary Conditions 6.

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Diffusion in Higher Dimensions 7. Subordinated Processes 8. Markov Processes and Semi-Groups 9. The Renewal Theorem 2.

## An Introduction to Probability & Statistics

Proof of the Renewal Theorem 3. Refinements 4. Persistent Renewal Processes 5. The Number N t of Renewal Epochs 6. Terminating Transient Processes 7. Diverse Applications 8. Existence of Limits in Stochastic Processes 9. Renewal Theory on the Whole Line Basic Concepts and Notations 2. Types of Random Walks 3. Distribution of Ladder Heights. Wiener-Hopf Factorization 3a. The Wiener-Hopf Integral Equation 4. Examples 5. Applications 6. A Combinatorial Lemma 7. Distribution of Ladder Epochs 8. The Arc Sine Laws 9. Miscellaneous Complements Tauberian Theorems.

Resolvents 1. The Continuity Theorem 2. Elementary Properties 3. Examples 4. Completely Monotone Functions. Inversion Formulas 5. Tauberian Theorems 6.

### Experiment, Sample Space and Event

Stable Distributions 7. Infinitely Divisible Distributions 8. Higher Dimensions 9. The course would be appropriate for seniors in mathematics or statistics or data science or computer science.

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It is also appropriate for first year graduate students in any It is also appropriate for first year graduate students in any of these fields. Content is up-to-date. In fact, the way simulations are used to illustrate important concepts in probability and statistics is now more relevant that ever! So, this is the right book or resource and No Need to Re-invent the Wheel!!!!

The book is very clear and smooth. Everything is classic or traditional except few places where I noticed a difference of what I am used to see: the authors used a unique notation, m x , for the distribution function cdf in the discrete case compared to that for the continuous case. Also,I am not sure that the selected vector and complement notations are commonly used.

The book is consistent and the material flows nicely! I love the connection made with other areas! I love the use of Paradoxes. Modularity is another major strength of the book! Although the material is nicely connected but but once can easily select to cover certain parts and skip others without creating gaps or difficulties in the students leering. The flow of the coverage and the nature of the probability area help in this matter.

You can easily treat or cover the discrete random variables separately and select the related material without any difficulties. You can do the same thing for the continuous case. You can leave some of the challenging examples that include some of the paradoxes that maybe challenging for students! You can also easily and smoothly teach or assign the history and development of the selected topics as reading s without making it as a part of the graded course!

Overall, the material is presented in a smooth way! Of course that is a matter of style, depending on the audience, I think it is easier to teach the material in Ch10 moment generating functions , then may be add a section about movements. I would probably slightly modify it. I would point at few other things later! The book is written with examples and problems that are very relevant to the culture we are in.

Examples form the business world examples include insurance coverage and insurance-related problems, gambling and lottery, sports, etc. Yes, I have specific comments that maybe useful to the authors: First: Thank you: Thank you for writing such a wonderful book. It is very clear, that the standards you held are really high and the timing of the book is unbelievable appropriate!

If you are interested, please let me know. The book covers the fundamentals of probability theory with quite a few practical engineering applications, which seems appropriate for engineering students to connect the theory to the practice. Each chapter contains realistic examples that apply Each chapter contains realistic examples that apply probability theory to basic statistical inference and naturally connect to the Monte Carlo simulations and graphical illustration of the probability distributions and probability density functions.

Students with basic calculus and discrete math can easily follow the development of probabilistic modeling and important properties of the popularly used probability distributions. Only the ergodic Markov chain and random walk appear challenging for the undergraduate students to comprehend without the formal introduction of stochastic process.

The book can serve as an introduction of the probability theory to engineering students and it supplements the continuous and discrete signals and systems course to provide a practical perspective of signal and noise, which is important for upper level courses such as the classic control theory and communication system design.

The material seems up-to-date and may be appealing to students with experience of Matlab to simulate various random events including the Markov chain and random walk covered in the later chapters. The book is well written with many interesting exercise problems for students to enhance their understanding. It would be nice to provide a few solutions to the selected problems. The discrete and continuous random variables and popularly used distributions can be summarized in separate tables e.

The book is well organized with coherent logical development. It would be nice to add a brief introduction to continuous time and discrete time stochastic processes before introducing the Markov chain and random walk. It contains many interesting examples to demonstrate how to apply a probabilistic modeling or statistical procedure to study the real world phenomena. The book consists of 12 chapters, 3 appendices with tables and index. It is designed for an introductory probability course, for use in a standard one-term course, in which both discrete and continuous probability is covered.

This book covers a This book covers a little bit more than I would normally cover in a probability class Markov chains and random walks and omits nothing that I would normally cover. All subject areas address in the Table of Contents are covered thoroughly. The book is mathematically accurate as far as I can tell. Examples are worked out in full detail throughout the text. In the earlier version were some mistakes, but have been corrected errata is available on the website.

All of these errors have been corrected in the current web version. The content is as up-to-date as any introductory probability textbook can reasonably be. In terms of longevity, the fact that the text of the book is stored in LaTeX ensures that the text will be useful for a long time to come. Updates will be straightforward to implement. There are over exercises in the text.

There are exercises to be done with and without the use of a computer and more theoretical exercises. A solution manual is available to instructors from website odd-numbered exercices or from the authors. In the text the computer is utilized in several ways: simulation, graphical illustration and to solve problems that do not lend to closed-form formulas. I think that the text in this book is extremely clear, which is great for a first course in probability.

It helps a large number of figures illustrating the discussed ideas. Authors have tried to present probability without too much formal mathematics but without sacrificing rigor. They have tried to develop the key ideas to provide a variety of interesting applications in normal live. The text is divided into small subsections with separate exercices for students to read there are easily be used as modules.

The organization is fine. The book presents all the topics in an appropriate sequence. I expect that instructors using this book would be using the material in the presented order maybe Combinatorics first.

### An introduction to probability theory and its applications.

The interface is OK. I didn't experience any problems. The lack of color graphics even in digital version few times authors use light blue color. I reviewed using the pdf version of the book. This does not have a linked table of contents, which would allow direct access to the sections. I wish the pdf file had this functionality.

## Introduction to Probability

The lack of hyperlinks is somewhat troublesome. I found no grammatical errors in this textbook but English is not my native language. It is very well written. No portion of this text appeared to me to be culturally insensitive or offensive in any way, shape, or form. I think that this textbook provides a great introduction to probability!

With such textbook available to students for free, I do not see any reasons to force my students to purchase a different textbook. My only complaint concerns the software. I would have preferred programs to be written in the language R. There are numerous very interesting historical comments in the text.

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continuedto influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarlyactivity from music to physics, and in daily experience from weather prediction topredicting the risks of new medical treatments.

This text is designed for an introductory probability course taken by sophomores,juniors, and seniors in mathematics, the physical and social sciences, engineering,and computer science. It presents a thorough treatment of probability ideas andtechniques necessary for a form understanding of the subject.

The text can be usedin a variety of course lengths, levels, and areas of emphasis. For use in a standard one-term course, in which both discrete and continuousprobability is covered, students should have taken as a prerequisite two terms ofcalculus, including an introduction to multiple integrals. In order to cover Chapter 11, which contains material on Markov chains, some knowledge of matrix theoryis necessary.

The text can also be used in a discrete probability course. The material has beenorganized in such a way that the discrete and continuous probability discussions arepresented in a separate, but parallel, manner. This organization dispels an overlyrigorous or formal view of probability and o? For use in a discrete probability course, studentsshould have taken one term of calculus as a prerequisite.

Very little computing background is assumed or necessary in order to obtain fullbenefits from the use of the computing material and examples in the text.