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ECMM450  Stochastic Processes (2019)
MODULE TITLE  Stochastic Processes  CREDIT VALUE  15 

MODULE CODE  ECMM450  MODULE CONVENER  Dr Christian Bick (Coordinator) 
DURATION: TERM  1  2  3 

DURATION: WEEKS  0  11  0 
Number of Students Taking Module (anticipated)  10 

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.
Prerequisite skills/knowledge: Discrete mathematics, probability theory, basic statistics
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.
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, problemsolving skills, timemanagement skills and facility to understand complex and abstract ideas.
 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 statedependent 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, nonqueueing 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;
 ChapmanKolmogorov 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.
Scheduled Learning & Teaching Activities  33.00  Guided Independent Study  117.00  Placement / Study Abroad  0.00 

Category  Hours of study time  Description 
Scheduled learning and teaching activities  33  Lectures/example classes 
Guided independent study  117  Guided independent study 
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  
Coursework  25  Written Exams  75  Practical Exams  0 

Form of Assessment  % of Credit  Size of Assessment (e.g. duration/length)  ILOs Assessed  Feedback Method 

Written exam – closed book  75  2 hours  Summer Exam Period  All  None 
Coursework  project  25  2000 words  All  Written 
Original Form of Assessment  Form of Reassessment  ILOs Reassessed  Time Scale for Reassessment 

All above  Written exam (100%)  All  August Referral/Deferral period 


If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.
If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.
information that you are expected to consult. Further guidance will be provided by the Module Convener
Basic reading:
ELE: http://vle.exeter.ac.uk/
Web based and Electronic Resources:
Other Resources:
Reading list for this module:
Type  Author  Title  Edition  Publisher  Year  ISBN  Search 

Set  Jones P.W. and Smith P.  Stochastic Processes: methods and applications  Arnold  2001  0000340806540  [Library]  
Set  Ross, Sheldon M  Introduction to Probability Models  10th  Elsevier  2010  9780123756862  [Library] 
CREDIT VALUE  15  ECTS VALUE  7.5 

PREREQUISITE MODULES  None 

COREQUISITE MODULES  None 
NQF LEVEL (FHEQ)  7  AVAILABLE AS DISTANCE LEARNING  No 

ORIGIN DATE  Tuesday 10 July 2018  LAST REVISION DATE  Wednesday 03 July 2019 
KEY WORDS SEARCH  Stochastic processes; probability models; Markov process. 
