# STOCHASTIC PROCESSES - 2024/5

Module code: MAT2003

## Module Overview

Stochastic processes are a series of random variables. Students will be introduced to both Markov Chains and continuous Markov stochastic processes, where the distributions of future random variables are determined by the value of the most recent random variable. These models are important for modelling things which change over time, such as voting intention or population size.

### Module provider

Mathematics & Physics

KUEH Audrey (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 58

Lecture Hours: 33

Tutorial Hours: 11

Guided Learning: 15

Captured Content: 33

Semester 2

None

## Module content

Indicative content includes:

• Probabilities and expectations of a Markov Chain;

• One-step and t-step transition probabilities of a Markov Chain;

• Gambler’s Ruin and other random walks;

• Properties of Markov Chains: recurrence/transience, periodicity and irreducibility;

• Basic Limit Theorem and stationary distributions;

• Poisson processes, pure birth processes and birth and death processes.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 20
Examination Exam (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:¿

• Interpretation and manipulation of mathematical statements.

• Subject knowledge through implicit recall of key definitions, theorems and their proofs.

• Analytical ability to calculate probabilities, expectations and long-term distributions.

Thus, the summative assessment for this module consists of:

• One in-semester test taken during the semester, worth 20% of the module mark, corresponds to Learning Outcome 1.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponds to Learning Outcomes 1, 2, 3, 4.

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Individual written feedback is provided to students for formative unassessed courseworks. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback during lectures and tutorials.

## Module aims

• Introduce students to Markov Chains and enable them to calculate simple probabilities and expectations.
• Facilitate students' understanding of the long-term behaviours of Markov Chains by studying their properties in a formal way.
• Enable students to determine long-term behaviour of a given Markov Chain.
• Introduce students to continuous Markov processes and relate them to differential equations.

## Learning outcomes

 Attributes Developed 001 Students will understand discrete and continuous Markov processes and calculate simple probabilities and expectations. KC 002 Students will be able to state and prove definitions and theorems about the long-term behaviour of discrete Markov processes. KC 003 Students will be able to find the long-term behaviour of a given discrete Markov process, occurring for example in politics. KCT 004 Students will be able to find the related differential equation of a given continuous Markov process, occurring for example in biology. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Give a detailed introduction to Markov Chains and continuous Markov processes and ensure experience in the methods used to interpret, understand, prove theorems and solve problems related to probabilities and expectations of stochastic processes.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• Eleven tutorials for guided discussion of solutions to problem sheets (provided to students in advance for completion to reinforce their understanding and guide their learning).

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students the opportunity to review parts of lectures that they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2003

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

• Digital Capabilities: The SurreyLearn page for MAT2003 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies and facilitates collaborative learning and information sharing.

• Employability: The module MAT2003 equips students with skills which significantly enhance their employability. Students gain mathematical proficiency, which hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

• Global and Cultural Capabilities: Students enrolled in MAT2003 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

• Resourcefulness and Resilience: MAT2003 is a module which demands a rigorous approach to Stochastic Processes, to which students will learn to adapt. They will gain skills in analysing problems and lateral thinking. Students will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.