Bachelor of Science (BSc)

BSc Mathematics with Placement Year

Strengthen your employability by taking our Placement Year programme.
  • Duration: 4 years
  • Year of entry: 2025
  • UCAS course code: G101 / Institution code: M20
  • Key features:
  • Industrial experience
  • Accredited course

Full entry requirementsHow to apply

Fees and funding

Fees

Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £34,500 per annum. For general information please see the undergraduate finance pages.

Policy on additional costs

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Course unit details:
Probability and Statistics 2

Course unit fact file
Unit code MATH27720
Credit rating 20
Unit level Level 2
Teaching period(s) Full year
Available as a free choice unit? No

Overview

The first part of this unit continues the development of Probability theory from Year 1 and provides an important basis for many later courses in Probability and Finance.   

The second part of this course unit provides students with the methodological foundations in model-based statistical learning, in particular likelihood estimation and inference and Bayesian learning. The theoretical and methodological discussions are complemented by practical computer application. 

Pre/co-requisites

Unit title Unit code Requirement type Description
Probability I MATH11711 Pre-Requisite Compulsory
Statistics I MATH11712 Pre-Requisite Compulsory

Aims

The first part of the course unit aims to develop a solid foundation in the calculus of probabilities and indicate the relevance and importance of this to tackling real-life problems. 

The second part aims to  

  • introduce the foundations of model-based statistical learning, 
  • introduce the general principles of likelihood-based inference and testing for general models (i.e. for both discrete and continuous distributions), 
  • offer a first overview of Bayesian statistical methods, and 
  • demonstrate corresponding computational procedures in R.

Learning outcomes

On the successful completion of the course, students will be able to:

  • calculate marginal distributions of and conditional distributions associated with multivariate random variables 
  • describe a range of parametric families to model their probability distribution 
  • calculate expectations and conditional expectations 
  • evaluate the distribution of functions of random variables 
  • use the properties of generating functions to derive moments of distributions, distributions of sums of random variables and limit distributions for sequences of random variables.
  • apply model-based approaches in statistical data analysis; 
  • derive maximum likelihood estimates and compute corresponding confidence intervals; 
  • perform statistical testing from a likelihood perspective; 
  • solve standard modelling and inference problems from a Bayesian point of view; 
  • use R to apply techniques from the course on actual data.  

Syllabus

Syllabus: 
Semester 1: 

Chapter 1: Random Variables (4 lectures) Definition of Events and their probabilities; definition of Random variables and their distributions; features of Discrete and Continuous random variables; Functions of random variables and mixed random variables. 

Chapter 2: Multivariate random variables (6 lectures) Bivariate Distributions; Independence; Sums of several variables; Conditional distributions; the bivariate transform. 

Chapter 3: Expectation (6 lectures) Expectation of a univariate random variable; Variance and higher Moments; Expectation of a bivariate random variable and conditional expectation; Probability generating functions; Moment generating functions; Sums of random variables using generating functions. 

Chapter 4: Sampling and convergence (6 lectures); The sample mean; Central limit theorem; Chebyshev's Inequality; Poisson Limit Theorem and characteristic functions; Introduction to the multivariate normal distribution. 

Semester 2: 

Section I - Likelihood 

- Entropy foundations: Shannon and differential entropy, cross-entropy, Kullback-Leibler (KL) divergence, expected Fisher information, minimum KL divergence and maximum likelihood. 

- Likelihood-based estimation: Likelihood function, regular models, score function, maximum likelihood estimators (MLE), invariance principle, relationship to ordinary least-squares estimation (OLS), observed Fisher information. 

- Quadratic approximation and normal asymptotics: Quadratic approximation of log-likelihood function and normal distribution, quantifying the uncertainty of MLEs using Fisher information, (squared) Wald statistic, normal confidence intervals, non-regular models. 

-  Likelihood-based inference: Likelihood-based confidence interval, Wilks log-likelihood ratio statistic, likelihood ratio test, generalised likelihood ratio test, optimality properties. 

Section II - Bayes 

- Conditioning and Bayes rule:  Conditional probability, Bayes’ theorem, conditional mean and variance, conditional entropy and chain rules, complete data log-likelihood, observed data log-likelihood, learning unobservable states using Bayes theorem. 

- Principles of Bayesian learning:  Prior and posterior probabilities and densities over parameters, marginal likelihood, sequential updates, summaries of posterior distributions and credible intervals, Bayesian and frequentist interpretation of probability. 

- Standard models: Beta-binomial model (for a proportion), normal-normal model (for the mean), inverse-gamma-normal model (for the variance), properties of Bayesian learning. 

- Bayesian model comparison: Log-marginal likelihood as penalised likelihood, model complexity, Bayes factor, Schwarz approximation and Bayesian Information Criterion (BIC), Bayesian testing using false discovery rate. 

- Choosing priors and optimality properties: default priors, uninformative priors, empirical Bayes, shrinkage estimation, James-Stein estimator, Frequentist properties of Bayesian estimators, optimality of Bayes inference (e.g. Cox theorem). 

Teaching and learning methods

This course unit forms the core of Theme C, and as such is delivered over 2 semesters.  Teaching is composed of two hours of lectures per week, and one tutorial class per fortnight.  Teaching materials will be uploaded to Blackboard for reference and review.

Assessment methods

Method Weight
Other 20%
Written exam 80%

Other: one mid-term online timed test and one mid-term written test, 40-50 mins each.

Feedback methods

Marked scripts within a week of the in-class test and automatically given feedback following the online test. Generic feedback made available after marks are released.

Recommended reading

Semester 1: 

Mood, A. M., Graybill, F. A. and Boes, D. C., Introduction to the Theory of Statistics, 3rd edition, McGraw-Hill 1974 

S. Ross, A First Course in Probability, 4th edition, Macmillan. 

D. Stirzaker, Elementary Probability, Cambridge University Press. Available electronically 

Neil A. Weiss, A Course in Probability, Pearson. 

Semester 2: 

Strimmer, K. 2024. MATH27720 Part 2 lecture notes. (Essential) 

Held, L, and Bove, D.S. 2020.  Applied Statistical Inference: Likelihood and Bayes (2nd edition). Springer (Recommended)  

Study hours

Scheduled activity hours
Lectures 44
Tutorials 12
Independent study hours
Independent study 144

Teaching staff

Staff member Role
Robert Gaunt Unit coordinator
Korbinian Strimmer Unit coordinator
Xiong Jin Unit coordinator

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