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ST452 Half Unit

Probability and Mathematical Statistics I

**This information is for the 2021/22 session.**

**Teacher responsible**

Prof Umut Cetin

**Availability**

This course is available on the MSc in Quantitative Methods for Risk Management. 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. Prior training on basic concepts of real analysis providing experience with formal proofs, sequences, continuity of functions, and calculus and is at the discretion of the instructor.

**Course content**

This course provides theoretical and axiomatic foundations of probability and mathematical statistics. 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 year, some of this teaching may be delivered through a combination of classes and flipped-lectures delivered as short online videos.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: 2 hours, reading time: 10 minutes) in the summer exam period.

Coursework (30%) in the MT.

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

**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.

**Important information in response to COVID-19**

Please note that during 2021/22 academic year some variation to teaching and learning activities may be required to respond to changes in public health advice and/or to account for the differing needs of students in attendance on campus and those who might be studying online. For example, this may involve changes to the mode of teaching delivery and/or the format or weighting of assessments. Changes will only be made if required and students will be notified about any changes to teaching or assessment plans at the earliest opportunity.

** Key facts **

Department: Statistics

Total students 2020/21: 2

Average class size 2020/21: 2

Controlled access 2020/21: No

Value: Half Unit

**Personal development skills**

- Problem solving
- Application of numeracy skills
- Specialist skills