### Understanding Markov Chains: Examples and Applications (Springer Undergraduate Mathematics Series)

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Many examples of metric spaces and topological spaces will be introduced and fundamental ideas within topology will be discussed, including separation axioms, compactness and connectedness. This module provides an introduction to the theory of stochastic processes and to their use as models, including applications to population processes and queues. The syllabus includes the Markov property, Chapman-Kolmogorov equations, classification of states of Markov chains, decomposition of chains, stationary distributions, random walks, branching processes, the Poisson process, birth-and-death processes and their transient behaviour, embedded chains, Markovian queues and hidden Markov models.

This module aims to show how the frequencies of characteristics in large natural populations can be explained using mathematical models and how statistical techniques may be used to investigate model validity. The syllabus includes: Mendel's First and Second Laws, random mating and random union of gametes, Hardy-Weinberg equilibrium, linkage, inbreeding, assortative mating, X-linked loci, selection and mutation.

This module is intended to offer a re-examination of standard statistical problems from a Bayesian viewpoint and an introduction to recently developed computational Bayes methods. The syllabus includes Bayes' theorem, inference for Normal samples; univariate Normal linear regression; principles of Bayesian computational, Markov chain Monte Carlo - theory and applications.

This module will study the practical analysis of spatial data. It commences with a discussion on different types of spatial data.

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Spatial point processes, random fields and spatial models for lattice data are discussed. There is a strong focus on the practical and computational aspects of model fitting and modern, computationally efficient model fitting software is introduced. The module introduces the concept of financial risk and discusses the importance of its regulation. The emphasis is laid on the popular risk measure Value at Risk VaR. After a brief discussion on asset returns, various modelling techniques - ranging from the simple Historical Simulation to the more advanced ARMA and GARCH models - are presented and applied for the calculation of VaR using real financial data.

The aim of this module is to provide a solid basis in risk management for those students considering a career in finance. Students are introduced to the application of mathematical models to financial instruments. The course will include an overview of financial markets and the terminology in common usage but the emphasis will be on the mathematical description of risk and return as a means of pricing contracts and options.

This module will explore real world applications of mathematics to biological problems e. The mathematical techniques used in the modelling will be nonlinear difference equations and ordinary differential equations. The module will be useful to students who wish to specialise in Applied Mathematics in their degree programme. The module will consider the mathematical and physical principles that describe the theory of electric and magnetic fields.

It will first describe the basic principles of electrostatics and magneto-statics and following this electrodynamics. Next Maxwell's equations are described along with the properties of electro-magnetic waves in a variety of media. Finally an application to the area of plasma physics is carried out through considering the orbits of charged particles in a variety of spatially and time varying magnetic fields.

The student will choose a project from a list published annually although a topic outwith the list may be approved. Students will be required to report regularly to their supervisor, produce a substantial written report, submitted by the end of April, and give a presentation.

## Understanding Markov Chains - Examples and Applications | Nicolas Privault | Springer

This module aims to show how the methods of estimation and hypothesis testing met in and level Statistics modules can be justified and derived; to extend those methods to a wider variety of situations. The syllabus includes: comparison of point estimators; the Rao-Blackwell Theorem; Fisher information and the Cramer-Rao lower bound; maximum likelihood estimation; theory of Generalized Linear Models; hypothesis-testing; confidence sets. The aims of this module are to introduce students to and interest them in the principles and methods of design-based inference, to convince them of the relevance and utility of the methods in a wide variety of real-world problems, and to give them experience in applying the principles and methods themselves.

By the end of the module students should be able to recognise good and poor survey design and analysis, to decide upon and implement the main types of survey design in relatively straightforward settings, and analyse the resulting survey data appropriately. The syllabus includes fundamentals of design based vs model-based inference, simple random sampling, sampling with replacement, ratio and regression estimators, stratified sampling, cluster sampling and unequal probability sampling.

This module introduces a wide range of features that occur in real comparative experiments. The applications include trials of potential new medicines by the pharmaceutical industry; comparisons of new varieties of wheat for bread-making; evaluating different machine settings in industry. Issues include whether and how to partition the experimental material into blocks for example, do old and young people respond to this drug differently? The module includes enough about the analysis of data from experiments to show what has to be considered at the design stage.

It also includes considerations of consultation with the scientist and interpretation of the results. Syllabus Introduction to concepts in the design of real comparative experiments. Randomization, replication, power. Simple linear model, orthogonal subspaces, analysis of variance.

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Fixed effects or random effects. Orthogonal designs. Factorial designs. Main effects and interactions. Control treatments. Row-column designs. Latin squares. Observational units smaller than experimental units. False replication. Split-plot designs.

## Understanding Markov Chains: Examples and Applications

Treatment effects in different strata. Structures defined by families of orthogonal factors. Eigenspaces of highly structured variance-covariance matrices. Showing factors on a Hasse diagram.

Using the Hasse diagram to calculate degrees of freedom and allocate treatment effects to strata. Skeleton analysis of variance. Cox, Planning of Experiments, Wiley The dissertation must consist of approximately 6, words of English prose on a topic agreed between the student and two appropriate members of staff who act as supervisors.

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The topic does not have to relate to work covered in previous Honours modules, though it may be helpful to the student if it builds on previous work. The topic and range of sources should be chosen in consultation with the supervisors in order to determine that the student has access to sources as well as a clear plan of preparation.

Available only to students in the second year of the Honours Programme, who have completed the Letter of Agreement. The aim of the project is to develop and foster the skills of experimental design, appropriate research management and analysis. The topic and area of research should be chosen in consultation with the supervisors in order to determine that the student has access to sources as well as a clear plan of preparation.

Letter of Agreement. This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples gambling processes and random walks are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters.

An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions. Understanding Markov Chains : Examples and Applications. Nicolas Privault. Probability Background.

Gambling Problems. DiscreteTime Markov Chains.