Build the probability space
Translate experiments into sample spaces, outcomes, and events.
Worcester Polytechnic Institute · instructor of record
MA 2631: Probability Theory
An axiomatic introduction to probability, from probability spaces, counting, conditioning, and independence to discrete, continuous, and joint distributions, expectation and covariance, transformations, and limit theorems.
Course approach
The course begins with concrete experiments—dice rolls, sports series, urn models, and the birthday problem—to make students identify the experiment, sample space, outcomes, and events before assigning probabilities. Particular attention is given to how the question being asked changes the appropriate sample space.
We then formalize these ideas using set operations and the axioms of probability. Rules such as complementation, inclusion–exclusion, conditional probability, the law of total probability, Bayes' theorem, and independence are developed from this foundation rather than introduced as unrelated formulas.
Develop random variables
Move from discrete and continuous distributions to joint behavior.
Study sums and limits
Use convolution, the central limit theorem, and the law of large numbers.
Representative course materials
syllabus.pdf
Syllabus
Course structure, policies, learning objectives, grading, and schedule.
lecture_notes.pdf
Lecture notes
A representative set of notes, worked examples, and course development.
exam_1.pdf
Sample Exam 1
A representative earlier-course assessment.
exam_2.pdf
Sample Exam 2
A representative later-course assessment.
Conceptual supplements
These short slide decks pause the main course sequence to develop ideas that benefit from a more visual, interpretive, or forward-looking treatment.
Interpretation
Why covariance and correlation measure linearity
Derives covariance and correlation through least-squares linear prediction, clarifying both what correlation measures and why nonlinear dependence can remain invisible to it.
Visualization
The Central Limit Theorem
Tracks sums of independent Uniform(0,1) variables from convolution to standardization, making the emergence of the Gaussian profile visible before stating the limiting theorem.
Stochastic dynamics
Introduction to Markov chains
Builds the transition from deterministic trajectories to probability distributions, then introduces the Markov property, transition matrices, and long-term state evolution.