## ST2PST-Probability and Statistical Theory

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:5
Terms in which taught: Autumn / Spring / Summer module
Pre-requisites: ST1PS Probability and Statistics and MA1FM Foundations of Mathematics
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA2PT1 Probability Theory I
Current from: 2019/0

Module Convenor: Dr Jeroen Wouters

Type of module:

Summary module description:
This module develops the theoretical foundations of methods used in statistical practice.
The module rigorously introduces basic concepts of probability from a mathematical perspective. It aims to equip the students with a basic knowledge in probability which will reveal the interplay between probability theory and fundamental areas of mathematics, will allow students to formulate general real or abstract problems in a probabilistic model and will unravel the fundamentals which statistical methods are built on. In more detail the module will be developed around the concepts of probability distributions, random variables, independence, sums of random variables, limit laws and their application (Central Limit Theorem and laws of large numbers), and structures that depend on the present to study the future evolution of stochastic phenomena (Markov chains). The module also covers key topics in estimation and statistical inference, including method of moments and maximum likelihood.

Aims:
This module aims to introduce students to some of the fundamental concepts and results of probability, and to some fundamental methods of statistical analysis. Many of these methods assume or are motivated by statistical models.

The module covers random variables together with probability distributions as the fundamental objects of probability theory, the concept of dependence/independence, which lead then to fundamental asymptotic results as well as a first introduction of stochastic processes such as Markov chains. There is a review and introduction of fundamental distributions such as chi squared, t and F distributions and relationships between them. The method of moments and the method of maximum likelihood are considered for point estimation of parameters, and properties of estimators, such as bias and mean square error, are described. Interval estimation and hypothesis testing are also developed.

Assessable learning outcomes:
By the end of the module the students are expected to be able to:
- Identify and demonstrate understanding of the main concepts and definitions in probability theory
- Without the help of notes to state all and prove some of the main results
- Identify and formulate problems in terms of probability and solve them to build up a simple stochastic model
- Use the main results to do various approximations
- Be familiar with a variety of mathematical techniques used in statistical inference
- Identify some commonly used distributions and the relationships between them
- Justify the use of, and apply, methods of estimation and state and derive the properties of estimators;
- Describe, justify and make use of the concepts of hypothesis testing and confidence intervals.

At the end of the module students will have some insight in the interrelation between other maths modules and probability and their relevance for applications.

Outline content:

Random variables with uniform distribution, continuous and discrete, distributions with densities and weights, expectation of random variables, the concept of independence, sums of independent random variables, concepts of convergence of random variables, dependent random variables and conditional distributions, Markov chains, graphical models and Bayesian statistics.

Standard distributions: review of common discrete distributions, normal, gamma, chi-squared, t-and F- distributions. Introduction to inference.

Point estimators: bias, mean square error, sufficiency, minimum variance unbiased estimators. Estimation methods: method of moments, maximum likelihood.

Confidence intervals: likelihood technique, central limit theorem. Hypothesis testing: basic principles; likelihood ratio test.

Brief description of teaching and learning methods:
Lectures, problem sheets and lecture-based tutorials.

Contact hours:
 Autumn Spring Summer Lectures 20 20 4 Tutorials 10 10 Guided independent study: 68 68 Total hours by term 98 98 4 Total hours for module 200

Summative Assessment Methods:
 Method Percentage Written exam 70 Set exercise 30

Summative assessment- Examinations:

3 hours

Summative assessment- Coursework and in-class tests:
Four assignments, and one examination paper.

Formative assessment methods:
Problem sheets

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:

One examination paper of 3 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous class test and coursework marks (70% exam, 30% coursework).