College
Statistical Theory
An original course in mathematical statistics: probability models generate data; sampling distributions justify estimation, uncertainty, testing and prediction. Derive likelihoods and estimators, examine theorem assumptions, build proofs and explore real simulations. Prerequisite: Probability Theory or equivalent undergraduate probability. Recommended: calculus, multivariable derivatives, probability distributions, joint laws, expectation, variance, LLN, CLT and mathematical reasoning. AP Statistics is not required. Includes 103 lessons with substantial written assessments and optional statistical-learning enrichment.
College · Upper Division · Advanced
The mathematics of randomness, built one idea at a time.
Construct probability models. Map outcomes to random variables. Derive distributions and expectations. Use calculus to analyze joint uncertainty, then explain why averages stabilize and standardized sums approach a normal law.
Your mathematical toolkit
Single-variable and multivariable calculus, algebra, infinite sums, and some experience with mathematical reasoning. Unit 0 refreshes the essentials. AP Statistics is not a prerequisite.
A semester of active learning
Plan for about 120 hours including reading, experiments, proofs, practice, and assessments. Work on paper alongside the labs. Random walks and Markov chains are optional enrichment.
Course journey
Follow the ideas as they grow
19 units, in the order they build on each other. Each unit's question is the thing it exists to answer.
0. Probability Theory readiness
Check the probability, calculus and reasoning that inference will rely on. Repair gaps in the existing Probability Theory course.
1. From populations to samples
Specify an identifiable probability model, distinguish data from random variables, and derive the meaning of a statistic.
2. Sampling distributions and order statistics
Derive exact and approximate laws for statistics, build pivots, and turn order-statistic events into binomial counts.
3. Point estimation and estimator quality
Evaluate estimators through bias, variance, squared-error risk, consistency, efficiency, and sensitivity to model departures.
4. Method of moments
Match empirical and theoretical moments, solve one- and multiple-parameter systems, and analyze validity and limitations.
5. Likelihood: evidence in a fixed sample
Construct likelihoods, understand probability versus likelihood, and use scores, support, and surfaces to interpret evidence.
6. Maximum likelihood estimation
Optimize common likelihoods, analyze resulting estimators, and handle constraints, invariance, numerical solutions, and pathologies.
7. Information and the Cramér–Rao bound
Turn likelihood sensitivity into a lower bound on unbiased estimation error, and find where the argument breaks.
8. Why estimators become approximately Normal
Connect consistency, Taylor expansion of the score, LLN, CLT, and the delta method.
9. Pivots and confidence procedures
Construct random intervals from probability statements, then distinguish coverage from posterior probability.
10. Tests, error probabilities, and p-value geometry
Design a decision rule, derive its error rates, and calculate a p-value from a specified null distribution.
11. Most powerful tests and their limits
Optimize power subject to a false-positive budget using likelihood ordering.
12. Constrained likelihood and likelihood-ratio tests
Compare the best fit under a restriction with the best fit in the full model.
13. Why counts produce chi-square inference
Derive Pearson discrepancies, fitted expected counts, and the geometry behind degrees of freedom.
14. Data compression without losing parameter information
Identify sufficient statistics and connect conditional distributions, likelihood, and estimator improvement.
15. Posterior inference and predictive uncertainty
Combine a stated prior with the sampling model, normalize, and distinguish posterior probability from repeated-sampling coverage.
16. Regression as a probability model
Derive least squares, its sampling distribution, and the assumptions behind coefficient inference.
17. Analysis of variance as a statistical model
Derive the decomposition and the F reference law, then connect group comparisons to regression.
18. Statistical learning · Enrichment
Connect binary-response likelihood and covariance geometry to prediction without replacing the inference core.