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 |
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Year | CATS Level | Semester | CATS Value |
Session 2021-22 | Level 6 FHEQ | First Semester | 15 |
Aims |
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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 |
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(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of: |
Syllabus |
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(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. (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 |
Recommended Texts |
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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 |
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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 |