Study information

Stochastic Processes - 2025 entry

MODULE TITLEStochastic Processes CREDIT VALUE15
MODULE CODEECMM450 MODULE CONVENERDr Kyle Wedgwood (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
Number of Students Taking Module (anticipated) 15
DESCRIPTION - summary of the module content
A stochastic process is one that involves random variables. A large number of practical systems within industry, commerce, finance, biology, nuclear physics and epidemiology can be described as stochastic and analysed using the techniques developed in this module. The systems considered may exist in any one of a finite, or possibly countably infinite, number of states. The state of a system may be examined continuously through time or at fixed and regular intervals of time.
 
You will study processes whose changes of state through time are governed by probabilistic laws, and you will learn how models of such processes can be applied in practice.
 
Pre-Requisite Modules:
None; but skills/knowledge of: Discrete mathematics, probability theory, basic statistics
AIMS - intentions of the module
 
The probability models considered in this module have a common thread running through them: that the behaviour of the system under consideration depends only on the state of the system at a particular point in time, and a probabilistic description of how the state of the system may change from one point in time to the next. The systems considered may exist in any one of a finite (or possibly countably infinite) number of possible states and the state of the system may be examined continuously through time or at fixed (and regular) intervals of time. A large number of practical systems within industry, commerce, finance, biology, nuclear physics and epidemiology, can be described and analysed using the techniques developed in this module.
INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge:

1. Demonstrate enhanced methodologies for tackling probabilistic problems;

2. Show awareness of a number of processes and systems whose behaviour through time are governed by probabilistic laws;

3. Construct and apply models describing that behaviour.

Discipline Specific Skills and Knowledge

4. Exhibit familiarity with the concept of random behaviour and the facility to analyse queues - skills which will be applied in later modules;

5. Display enhanced facility with the fundamental mathematical techniques of finite and infinite summation, and of differential and integral calculus.

Personal and Key Transferable / Employment Skills and Knowledge:

6. Reveal enhanced analytical skills, numerical skills, reasoning skills, problem-solving skills, time-management skills and facility to understand complex and abstract ideas.

SYLLABUS PLAN - summary of the structure and academic content of the module
Probability generating functions (PGFs): definition, basic properties and illustrative examples of PGFs;
- moments of random sums of random variables;
- branching processes: definition, PGF and moments of the population in generation n of a branching process;
- probability of ultimate extinction;
- stochastic size of original population;
- Poisson processes: definition;
- memoryless property;
- Erlang distribution of time to the nth event;
- Poisson distribution of number of events in a given period of time;
- binomial distribution of number r of events in t given n in T;
- beta distribution of time t to rth event given n events in T;
- combining and decomposing independent Poisson processes;
- queueing theory: differential equations for the transient behaviour of models with state-dependent Markov arrival and departure processes;
- derivation of the steady state behaviour of this model;
- existence conditions for steady state;
- specific queueing models: fixed arrival rate, finite source population, customer baulking behaviour, one or more servers, finite system capacity, non-queueing systems which can be modelled as queues;
- mean number of customers in the system/queueing;
- mean time spent in the system/queueing;
- statement and proof of Little's formula;
- distribution of time spent in system/queueing given first come first served;
- Markov processes: Markov property;
- time homogeneity;
- stochastic matrices;
- Chapman-Kolmogorov equations;
- classification of states: accessible, communicating, transient, recurrent, periodic, aperiodic;
- Ergodic Markov chains;
- renewal theorem;
- mean recurrence time;
- necessary/sufficient conditions for the system to tend to a steady state;
- random walks: definition of a random walk with absorbing/reflecting/elastic barriers;
- statement of, solution for and mean time to finish for the Gambler's Ruin problem.
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 22 Guided Independent Study 128 Placement / Study Abroad 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 22 Lectures/example classes
Guided independent study 128 Reflection on scheduled learning and teaching, further reading, preparation for assessments

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – example sheets     Tutorial sessions during lectures/office hours, written feedback on work

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Problem sheets 20   All Written
Exam (closed book) 80 2 hours All Written/verbal on request

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Problem sheets Problem sheets (20%) All August Referral/Deferral period
Exam (closed book) Exam (closed book) (2 hours, 80%) All Auust Referral/Deferral period

 

RE-ASSESSMENT NOTES
Reassessment will be by coursework and/or written exam in the failed or deferred element only. For referred candidates, the module mark will be capped at 50%. For deferred candidates, the module mark will be uncapped.
 
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener
Basic reading:
 
• Jones P.W. and Smith P., Stochastic Processes: methods and applications, Arnold, 2001, 000-0-340-80654-0
• Ross, Sheldon M, Introduction to Probability Models, 10th, Elsevier, 2010, 978-0123756862
 
Web-based and electronic resources: 
 
• ELE.

 

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Jones P.W. and Smith P. Stochastic Processes: methods and applications Arnold 2001 000-0-340-80654-0
Set Ross, Sheldon M Introduction to Probability Models 10th Elsevier 2010 978-0123756862
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES None
CO-REQUISITE MODULES None
NQF LEVEL (FHEQ) 7 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Thursday 14th March 2024 LAST REVISION DATE Wednesday 30th July 2025
KEY WORDS SEARCH Stochastic processes; probability models; Markov process.

Please note that all modules are subject to change, please get in touch if you have any questions about this module.