Introduction To Probability
An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. The 2nd edition is a substantial revision of the 1st edition, involving a reorganization of old material and the addition of new material. The length of the book has increased by about 25 percent. The main new feature of the 2nd edition is thorough introduction to Bayesian and classical statistics.
The book is the currently used textbook for "Probabilistic Systems Analysis," an introductory probability course at the Massachusetts Institute of Technology, attended by a large number of undergraduate and graduate students. The book covers the fundamentals of probability theory (probabilistic models, discrete and continuous random variables, multiple random variables, and limit theorems), which are typically part of a first course on the subject, as well as the fundamental concepts and methods of statistical inference, both Bayesian and classical. It also contains, a number of more advanced topics, from which an instructor can choose to match the goals of a particular course. These topics include transforms, sums of random variables, a fairly detailed introduction to Bernoulli, Poisson, and Markov processes.
The book strikes a balance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just intuitively explained in the text, but is developed in detail (at the level of advanced calculus) in the numerous solved theoretical problems.
Written by two professors of the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology, and members of the prestigious US National Academy of Engineering, the book has been widely adopted for classroom use in introductory probability courses within the USA and abroad.
From a Review of the 1st Edition:
...it trains the intuition to acquire probabilistic feeling. This book explains every single concept it enunciates. This is its main strength, deep explanation, and not just examples that happen to explain. Bertsekas and Tsitsiklis leave nothing to chance. The probability to misinterpret a concept or not understand it is just... zero. Numerous examples, figures, and end-of-chapter problems strengthen the understanding. Also of invaluable help is the book's web site, where solutions to the problems can be found-as well as much more information pertaining to probability, and also more problem sets. --Vladimir Botchev, Analog Dialogue
Several other reviews can be found in the listing of the first edition of this book. Contents, preface, and more info at publisher's website (Athena Scientific, athenasc com)
Why Read This Book
You will get a clear, mathematically rigorous foundation in probability and stochastic processes that directly underpins modern DSP, communications, radar, and audio/speech algorithms. The book balances intuition and precise proofs, and the expanded 2nd edition adds practical Bayesian and classical statistical tools you'll use for estimation, detection, and spectral analysis.
Who Will Benefit
Engineers and graduate students in DSP, communications, radar, and audio/speech processing who need a solid probabilistic foundation to model noise, analyze algorithms, and design estimators and detectors.
Level: Intermediate — Prerequisites: Single-variable calculus and basic linear algebra; familiarity with basic calculus-based probability ideas is helpful but not required.
Key Takeaways
- Model random signals and noise using probability distributions and stochastic processes, enabling principled analysis of DSP algorithms.
- Compute and manipulate expectations, moments, characteristic and generating functions to derive statistical properties of signals and filters.
- Apply limit theorems (Law of Large Numbers, Central Limit Theorem) to justify averaging, convergence, and statistical performance in spectral estimates and algorithms.
- Use Bayesian and classical statistical methods for parameter estimation, hypothesis testing, and detector design in communications and radar.
- Analyze Markov, Poisson, and other stochastic processes relevant to queueing, channel models, and event-driven signal phenomena.
Topics Covered
- 1. Introduction and Axioms of Probability
- 2. Counting, Combinatorics, and Discrete Distributions
- 3. Conditional Probability and Independence
- 4. Random Variables and Continuous Distributions
- 5. Expectation, Moments, and Inequalities
- 6. Generating and Characteristic Functions
- 7. Joint Distributions and Multivariate Methods
- 8. Convergence of Random Variables, LLN and CLT
- 9. Stochastic Processes: Definitions and Stationarity
- 10. Markov Chains and Poisson Processes
- 11. Introduction to Spectral Methods and Stationary Processes
- 12. Bayesian and Classical Statistical Inference
- 13. Estimation, Confidence Intervals, and Hypothesis Testing
- 14. Appendices: Useful Integrals and Mathematical Tools
Languages, Platforms & Tools
How It Compares
Compared with Ross's A First Course in Probability, Bertsekas offers more depth and intuitive rigor; compared with Papoulis's Probability, Random Variables, and Stochastic Processes, it is more focused on probabilistic foundations and modern statistical inference though less DSP-centric.
