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ST552 Half Unit

Probability and Mathematical Statistics I

**This information is for the 2022/23 session.**

**Teacher responsible**

Prof Umut Cetin

**Availability**

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.

**Teaching**

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**

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

**Assessment**

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

**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