Please note: there may be some adjustments to the teaching arrangements published in the course catalogue for 2020-21. Given current circumstances related to the Covid-19 pandemic it is anticipated that some usual arrangements for teaching on campus will be modified to ensure the safety and wellbeing of students and staff on campus; further adjustments may also be necessary, or beneficial, during the course of the academic year as national requirements relating to management of the pandemic are revised.

## Principles of Probability and Statistics (Level M) STATS5022

• School: School of Mathematics and Statistics
• Credits: 10
• Level: Level 5 (SCQF level 11)
• Typically Offered: Semester 1
• Available to Visiting Students: Yes
• Available to Erasmus Students: Yes

### Short Description

This course establishes crucial concepts and asymptotic results that are widely relied upon in probability and statistics. These include convergence in distribution and the Central Limit Theorem for probability, and optimal properties of estimators, particularly maximum-likelihood estimators, in statistics. It also provides an opportunity for students to read further into one topic related to the course and write an essay to summarise what they have learnt.

### Timetable

20 lectures (2 each week)

5 tutorials (fortnightly throughout the semester)

### Requirements of Entry

Some optional courses may be constrained by space and entry to these is not guaranteed unless you are in a programme for which this is a compulsory course.

### Excluded Courses

STATS4047 Principles of Probability and Statistics

### Assessment

120-minute, end-of-course examination (100%)

Main Assessment In: April/May

### Course Aims

The aims of this course are:

■ to introduce and discuss the concept of convergence in the theory of random variables;

■ to establish the laws of large numbers and the Central Limit Theorem;

■ to discuss optimal properties of point estimators;

■ to establish the large-sample properties of maximum-likelihood estimation;

■ to introduce students to the EM algorithm.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

■ describe and contrast convergence in probability, convergence in distribution, convergence in quadratic mean and almost sure convergence;

■ state, use and prove various probabilistic inequalities;

■ state, prove and use the Weak Law of Large Numbers, the Strong Law of Large Numbers and the Central Limit Theorem;

■ state and discuss optimal properties of point estimators;

■ state, prove and use the Rao-Blackwell Theorem and the Cramer-Rao lower bound;

■ state, prove and apply general asymptotic properties of maximum-likelihood estimators;

■ construct an EM algorithm for various missing data problems.

■ prove Hoeffding's inequality

■ write down the expression for the best linear unbiased estimator (BLUE) and explain how it is derived..

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.