Chapter 1: Course Introduction and Embracing Uncertainty
Meaning of probability, expected value, decision-making, forecasting and incentives
Chapter 2: Events, Axioms, and Probability Models
Sample spaces, events, set operations, probability axioms, basic identities
Chapter 3: Counting Principles and Equally Likely Outcomes
Product rule, permutations, combinations, counting models, equally likely outcomes
Chapter 4: Conditional Probability and the Law of Total Probability
Conditional probability, multiplication rule, partitions, law of total probability
Chapter 5: Bayes’ Theorem
Bayes’ rule, prior and posterior, inference using partitions
Chapter 6: Independence of Events
Independence, conditional independence, complements, pairwise vs mutual independence
Chapter 7: Random Variables
Random variables as mappings, inverse images, discrete variables, PMF
Chapter 8: Expectation and Variance (Discrete)
Expectation, linearity, variance, basic properties
Chapter 9: Inequalities and Confidence
Markov and Chebyshev inequalities, confidence bounds
Chapter 10: Bernoulli and Binomial Distributions
Bernoulli trials, binomial model, parameter interpretation
Chapter 11: Geometric and Negative Binomial Distributions
Waiting time models, memoryless property, repeated trials
Chapter 12: Poisson Distribution and Binomial Limit
Poisson model, rare events, binomial approximation
Chapter 13: Reliability and Union Bounds
System reliability, bounds, approximation methods
Chapter 14: Decision Making and Likelihoods
Likelihood functions, decision rules, estimation principles
Chapter 15: Hypothesis Testing (Discrete)
Testing framework, Type I/II errors, decision thresholds
References
[1] Bruce Hajek.
Probability with Engineering Applications: ECE 313 Course Notes.
January 2025, University of Illinois Urbana–Champaign.
[Link].
[2] Abhishek Umrawal.
Probability with Engineering Applications: ECE 313/MATH 362 Lecture Notes.
Fall 2025, University of Illinois Urbana–Champaign.
[Link].
License
© 2026 Abhishek Umrawal
This work is licensed under the Creative Commons Attribution–NoDerivatives 4.0 International License (CC BY–ND 4.0).
[View License].
Acknowledgments
This work is supported by the Strategic Instructional Innovations Program (SIIP),
The Grainger College of Engineering, University of Illinois Urbana–Champaign.
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