ST552      Half Unit
Probability and Mathematical Statistics I

This information is for the 2022/23 session.

Teacher responsible

Prof Umut Cetin


This course is available on the MPhil/PhD in Statistics. This course is available with permission as an outside option to students on other programmes where regulations permit.

The availability as an outside option requires a demonstration of sufficient background in mathematics and statistics and is at the discretion of the instructor.

Course content

This course provides theoretical and axiomatic foundations of probability and mathematical statistics, and is intended for PhD students in the Statistics department. In particular, the following topics will be covered:

1. Measure spaces; Caratheodory extension theorem; Borel-Cantelli lemmas.

2. Random variables; monotone-class theorem; different kinds of convergence.

3. Kolmogorov’s 0-1 law; construction of Lebesgue integral.

4. Monotone convergence theorem; Fatou's lemmas; dominated convergence theorem.

5. Expectation; L^p spaces; uniform integrability.

6. Characteristic functions; Levy inversion formula; Levy convergence theorem; CLT.

7. Principle and basis for statistical inference: populations and samples, decision theory, basic measures for estimators.

8. Estimation: U and V statistics, unbiased estimators, MVUE, MLE.

9. Hypothesis testing: Neyman-Pearson lemma, UMP, confidence sets.

10. Product measures; conditional expectation.


This course will be delivered through a combination of classes, lectures and Q&A sessions totalling a minimum of 30 hours across Michaelmas Term.  This course includes a reading week in Week 6 of Michaelmas Term.

Formative coursework

Students will be expected to produce 9 problem sets in the MT.

Weekly problem sets that are discussed in subsequent seminars. The coursework that will be used for summative assessment will be chosen from a subset of these problems.

Indicative reading

  1. Williams, D. (1991). Probability with Martingales. Cambridge University Press.
  2. Durrett, R. (2019). Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics.
  3. Shao, J. (2007). Mathematical Statistics. Springer Texts in Statistics.
  4. Keener, R. (2010). Theoretical Statistics. Springer Texts in Statistics.


Exam (70%, duration: 3 hours, reading time: 10 minutes) in the summer exam period.
Coursework (30%).

Three of the homework problem sets will be submitted and marked as assessed coursework.

Key facts

Department: Statistics

Total students 2021/22: 6

Average class size 2021/22: 5

Value: Half Unit

Guidelines for interpreting course guide information

Course selection videos

Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.

Personal development skills

  • Problem solving
  • Application of numeracy skills
  • Specialist skills