Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title APPLIED PROBABILITY
Code MATH362
Coordinator Dr E Azmoodeh
Mathematical Sciences
Ehsan.Azmoodeh@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2021-22 Level 6 FHEQ First Semester 15

Aims

To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ``dynamic" events occurring over time. To familiarise students with an important area of probability modelling.


Learning Outcomes

(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of:
(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes
(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain
(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.
2. Intellectual Abilities After the module, students should be able to:
(a) formulate appropriate situations as probability models: random processes
(b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems
3. General Transferable Skills
(a) numeracy through manipulation and interpretation of datasets
(b) communication through presentation of written work and preparation of diagrams
(c) problem solving through tasks set in tutorials
(d) time management in the completion of practicals and the submission of assessed work
(e) choosing, applying and interpreting results of probability techniques for a range of different problems.


Syllabus

 

(1) Introduction and preliminaries: Sample space, random variables, distribution functions. Conditional probabilities and expectations: definitions and properties; computing expectation by conditioning (discrete and continuous cases), computing probability by conditioning.

(2) Random walks: symmetric and asymmetric RWs, random walk with absorbing boundary: Gambler's ruin.

(3) Discrete time Markov chains: definition and examples, transition probabilities and matrices. Examples: weather model etc.
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(4) Higher order transition probabilities, Chapman Kolmogorov equations.

(5) Communication of states, periodicity, recurrence and transience.

(6) Asymptotic behaviour of Markov chains, limiting and stationary distributions. Absorbing probability.

(7) Exponential distribution, memoryless property, first failure, minimum of exponential random variables.

(8) Poisson processes: definitions and examples. Interarrival time and waiting time distributions, superposition.

(9) Poisson-like processes: compound Poisson process, non-stationary Poisson process, a selection of examples including short term insurances models.

(10) Markov Jump process: transition matrix, Chapman Kolmogorov Equations, Kolmogorov forward/backward eq., birth and
death process, M/M/1 queue, non-homogeneous Markov-process.


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH264 STATISTICAL THEORY AND METHODS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH162 INTRODUCTION TO STATISTICS 

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment on campus  60 minutes    50       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class Test  around 60-90 minutes    50