Introduction to Probability, Statistics, and Random Processes
This book introduces students to probability, statistics, and stochastic processes. It can be used by both students and practitioners in engineering, various sciences, finance, and other related fields. It provides a clear and intuitive approach to these topics while maintaining mathematical accuracy.
The book covers:
- Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods
- Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities
- Limit theorems and convergence
- Introduction to Bayesian and classical statistics
- Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion
- Simulation using MATLAB and R (online chapters)
The book contains a large number of solved exercises. The dependency between different sections of this book has been kept to a minimum in order to provide maximum flexibility to instructors and to make the book easy to read for students. Examples of applications—such as engineering, finance, everyday life, etc.—are included to aid in motivating the subject. The digital version of the book, as well as additional materials such as videos, is available at www.probabilitycourse.com.
Why Read This Book
You will gain a clear, mathematically sound foundation in probability, statistics, and stochastic processes tailored to engineering applications — from noise modeling to spectral analysis. The book balances intuition and rigor so you can apply probabilistic tools directly to DSP, communications, and radar problems without getting lost in unnecessarily abstract theory.
Who Will Benefit
Advanced undergraduates, graduate students, and practicing engineers in DSP, communications, radar, and related fields who need a practical, rigorous treatment of randomness and stochastic modeling.
Level: Intermediate — Prerequisites: Single-variable calculus, basic linear algebra, and comfort with calculus-based mathematics; no prior formal course in probability is required.
Key Takeaways
- Apply probability axioms and conditional reasoning to model random phenomena in signals and systems.
- Compute and use moment-generating and characteristic functions to derive distributions and moments.
- Use limit theorems (law of large numbers, central limit theorem) to justify approximations and sampling behaviors.
- Model and analyze random processes: characterize stationarity, autocorrelation, and power spectral density for spectral analysis.
- Formulate and solve basic estimation and detection problems using classical and Bayesian approaches (MLE, MAP, hypothesis testing).
- Analyze linear systems driven by stochastic inputs and compute output statistics (e.g., output PSD, SNR) useful for filter and system design.
Topics Covered
- 1. Introduction to Probability and Random Experiments
- 2. Axioms of Probability, Conditional Probability, and Counting Methods
- 3. Discrete and Continuous Random Variables
- 4. Multiple Random Variables and Joint Distributions
- 5. Expectation, Moments, Inequalities, and Transform Methods
- 6. Moment-Generating and Characteristic Functions
- 7. Limit Theorems and Modes of Convergence
- 8. Introduction to Bayesian and Classical Statistics (estimation and hypothesis testing)
- 9. Random Processes: Definitions, Mean, and Correlation
- 10. Stationarity, Ergodicity, and Power Spectral Density
- 11. Gaussian Processes, Noise Models, and Linear Systems with Random Inputs
- 12. Markov Processes, Poisson Processes, and Renewal Theory
- 13. Applications and Examples in Communications, Radar, and Signal Processing
- Appendices: Mathematical Tools and Solutions to Selected Problems
Languages, Platforms & Tools
How It Compares
Covers material similar to Papoulis' 'Probability, Random Variables, and Stochastic Processes' but in a more accessible, modern pedagogical style; for DSP-focused estimation and detection, it pairs well with Steven M. Kay's 'Fundamentals of Statistical Signal Processing'.
