ST2PST: Probability and Statistical Theory

Module code: ST2PST

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 2 (Intermediate)

When you’ll be taught: Semester 2

Module convenor: Dr Jeroen Wouters, email: j.wouters@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE ST1PS AND TAKE MA1FM ) OR ( TAKE ST1PSNU AND TAKE MA0FMNU ) (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded:

Placement information: NA

Academic year: 2024/5

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

This module rigorously introduces basic concepts of probability from a mathematical perspective and develops the theoretical foundations of methods used in statistical practice and data science applications. 

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. 

The module covers random variables together with probability distributions as the fundamental objects of probability theory, limit laws, as well as a first introduction of stochastic processes such as Markov chains. 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. 

Module learning outcomes

By the end of the module, it is expected that students will be able to: 

1. Identify and demonstrate understanding of the main concepts and definitions in probability theory 
2. Identify and formulate problems in terms of probability and solve them by constructing simple stochastic model 
3. Justify the use of, and apply, methods of estimation and state and derive the properties of estimators; 
4. Describe, justify and make use of the concepts of hypothesis testing and confidence intervals. 

Module content

Random variables with continuous, discrete and mixed distributions, expectation of random variables, independence, sums of independent random variables, generating functions, concepts of convergence of random variables, dependent random variables, conditional distributions, and Markov chains. 

Point estimators: Introduction to inference. Bias, mean square error, sufficiency, minimum variance unbiased estimators. Estimation methods: method of moments, maximum likelihood. 

Confidence intervals, likelihood technique, central limit theorem. Principles of hypothesis testing and likelihood ratio test. Introduction to Bayesian statistics. 

Structure

Teaching and learning methods

Lectures, supported by non-assessed problem sheets, weekly tutorials, and exercises. 

Study hours

At least 54 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.