MA8704 Probability Theory / ST3201 Master Seminar in Statistics – Spring 2026
Lectures Tuesday and Thursday 10:15-12 in Simastuen (656 SBII).
Week 2: General overview. Axioms for Events, Measurable sets, Open sets. Measure and probability. Random variable. Kolmogorov book
Week 3: Distribution, Expectation, Integral, Change-of-variables. Linearity. Limits Bahadur's theorem and Rudin book
Week 4: NO LECTURES Solve problems that will be here. You may use Simastuen lecture times for group work. The problems are representative for the kind of questions you may meet at the exam. Send handwritten solutions to me and you may get some feedback.
Week 5: Limit theorems for integrals. Problems and view towards statistics. mu and conditional. Quasi-integral
Week 6: Fubini and independence. Conditional distributions and statistics
Week 7: Conditional distributions and statistics. Smakebiter fra forelesning 10 (uferdig kladd): del 1 og del 2
Week 8: Proving Radon-Nikodym, Conditional expectation properties, statistics. F11 og F12 (uferdig kladd)
Week 9: Tuesday: Statistics and conditional distributions overview: Bayes, sufficiency, Basu, Lehmann-Scheffe, Halmos-Savage, Rao-Blacwell, Bahadur. F13. Thursday: Calculation of conditional. F14
Week 10: Convergence of measures and random variables. Karr: Ch 5. F15 og F16
Week 11: Convergence of measures and random variables. Independence and Bernoulli. Karr: Ch 5 + Ch 3.
Week 12: Characteristic Functions. Karr: Ch 6.
Week 13: Classical Limit Theorems. Karr: Ch 7.
Week 14: Easter. No lectures.
Week 15: Classical Limit Theorems. Karr: Ch 7.
Week 16: Martingales. Karr: Ch 9.
Week 17: Summary and questions lecture on Tuesday.
The course gives a broad introduction to classical probability theory and asymptotic techniques with a view towards applications in statistics. In particular, it provides mathematical definitions of expectation, conditional expectation, data, statistic, model, and parameter.
These basic concepts are illustrated in the textbooks used in the introductory course TMA4240 Statistics and the more advanced course TMA4295 Statistical Inference, but mathematical definitions are lacking.
An advantage of the elementary approach in such courses is that it can place more emphasis on intuitive understanding. However, there are also disadvantages. One is the appearance of seemingly paradoxical phenomena such as the Borel paradox. Another is that proofs that are simple in a general measure-theoretic framework may become more cumbersome and opaque in an elementary setting. An example is the 17-line proof of Basu’s theorem (restricted to discrete distributions) in the textbook used in TMA4295 Statistical Inference, whereas Basu’s original proof in full generality fits on a single line.
The course is well suited for students interested in a rigorous mathematical formulation of probability as needed in statistics. The level is comparable to that of the courses TMA4145 Linear Methods and TMA4225 Foundations of Analysis. It is advantageous, but not required, to have taken these courses.
First meeting: Tuesday 6 January 2026, 10:15–12:00 Room: 656, SBII
If you are interested in participating, you are kindly asked to send an email to: Gunnar.Taraldsen@ntnu.no
It would also be helpful if you could fill out the form: https://nettskjema.no/a/575834
Teaching material
Lecture notes (see above lecture plan!) and main book:
Allan F.Karr. Probability 1993. Springer Verlag. (Can be downloaded freely from Springer using an NTNU account.)
The following two lists of corrections are from the history of the course at NTNU and not correct. The first claimed corrections in each list are in fact wrong - the book is correct. I have not checked the other claimed 'corrections', so read with care! I have not found any direct errors in the book, so fewer errors than in most books I've seen on this topic.
Corrections to the main textbook.
Exam
Oral. Grade: Passed/Failed
TENTATIVE syllabus is given by lectures (see lecture notes) and Karr p.1-242 (Intro + Ch 1-8) + Ch 9 ?.