STOR602: Probability and Stochastic Processes

Credits: 20

Tutors: Azadeh Khaleghi, Dave Worthington, Peter Jacko and Kevin Glazebrook.

Outline: The module will provide an introduction to probability, random variables, Markov processes, Poisson processes and a range of key stochastic processes for modelling in statistics and operational research. The emphasis in the module will be on the evaluation of complex stochastic properties via both analytical methods and simulation. It will cover:

  • Introduction to probability building from the axioms.
  • Univariate random variables: standard distributions and their justifications, inter-relations, and properties.
  • Multivariate random variables: marginals and copulas, their decomposition as a series of conditionals, dependence measures, and standard distributions.
  • Simulation of random variables and approximation of their properties by Monte Carlo.
  • Transformations of random variables.
  • Poisson/Counting Processes.
  • Markov processes: discrete and continuous time, covering stationarity, conditional independence, and standard examples.
  • Spatial processes: Gaussian processes and their correlation functions.
  • Queueing models: time dependent and steady state behaviour, standard processes.
  • Replacement, inspection and maintenance models: assessment of different strategies.
  • Simulation of a range of stochastic processes to approximate properties.

Objectives: The key models in statistics and operational research are based on probabilistic theory for random variables and stochastic processes. By providing this probabilistic theory and studying a wide range of models, this module provides the building blocks for modelling and understanding the properties of all stochastic behaviour studied in the MRes. 

This module builds from the axioms of probability through the study of univariate and multivariate random variables to a range of stochastic processes and explores how a range of strategies for interaction with these processes affects their behaviour. The emphasis will differ from standard modules on this topic. Here a much greater focus will be on the importance of simulation methods to approximate properties, which is particularly valuable as in many cases analytic methods give intractable solutions.

Learning Outcomes: On successful completion of this module students will be able to:

  • recognise the contexts when certain random variables occur;
  • simulate a range of random variables;
  • derive a range of properties for random variables using analytic and simulation methods;
  • transform random variables;
  • demonstrate the effects of covariates and latent variables on the distribution of random variables;
  • identify a range of stochastic models and derive a range of their behaviours including Poisson/Counting Processes, Markov processes, Spatial processes, Queueing models.
  • simulate a range of stochastic processes;
  • use models of this type to assess different strategies for replacement, inspection and maintenance in a range of applications;
  • use R, EXCEL and specialist software to tackle a range of problems with stochastic processes which analytical methods cannot give tractable answers to.

Core texts:

  • Grimmett, G  and Stirzaker, D. R. (2001). Probability and Random Processes (3rd edition) OUP Oxford.
  • Ross, S. (2005). A First Course in Probability (Seven edition) Pearson Education.
  • Morgan, B.J. T. (1984). Elements of Simulation. Chapman and Hall.
  • Winston, W. L. (2004). Operations Research : Applications & Algorithms. Thomson/Brooks/Cole.
  • Kemeny, J. G and Snell, J. L. (1963). Finite Markov Chains. Princeton, N.J. : Van Nostrand.
  • Gross, D. and Harris, C. M. (1985) Fundamentals of Queueing Theory. Wiley.

Assessment: Assessment will be through a combination of summer exam (80%) and coursework (20%). The coursework consists of weekly exercises.  

Contact hours: There will be a mixture of lectures, tutorials and computer workshops totalling approximately 40 hours contact. In addition, private study will make up the majority of the learning hours.

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