Statistics
ST412: Stochastic Processes (5 ECTS)
RUNNING IN ACADEMIC YEAR 2016-2017.
The goal of the course is to introduce the main ideas and methods of stochastic processes with the focus on Markov chains (processes with discrete time index and finite state space). Branching processes and Poisson process (continuous time and discrete state space) will also be included in the study.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 28 hours (24 Lecture hours, 4 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Use probability and moment generating functions to calculate corresponding distributional properties.
- Derive properties of branching processes such as expectation, variance, and probability of extinction.
- Calculate relevant probabilities in random walks with and without barriers
- Use Markov property to prove various probabilistic statements about Markov chain
- Classify states of Markov chains and determine stationarity properties
- Calculate limiting and statitonary distributions
- Prove and calculate various properties of Poisson process
- Build and describe Markov chains to represent simplified real world problems, for example, such as those those used to model credit mobility
Indicative Content
The goal of the course is to introduce the main ideas and methods of stochastic processes with the focus on Markov chains (processes with discrete time index and finite state space). The topics include:
(i) A review of probability theory: Discrete and continuos random variables (r.v.), joint and conditional distributions, expectations, variance, sums of iid r.v.-s, conditional expectation;
(ii) probability generating functions, moment generating functions
(iii) Random sums of r.v.-s;
(iv) Branching processes;
(v) Markov property and Markov chains (MC): Random walk with absorbing barriers (Gambler’s ruin); Classification of states for a finite discrete Markov chain; Stationary and limiting prob. distribution of Markov chains; Random walk in 2 and more dimensions; Mean first passage times;
(vi) Poisson process (independent increments formulation; inter-arrival times formulation);
(vii) Applications of stochastic processes in finance, bioinformatics, computer science.
Module Resources
1) Hoel, Port, Stone, Stochastic Processes, Houghton & Mifflin
2) C. Grinstead and L. Snell, Introduction to Probability, American Mathematical Society (free online copy)
3) Sheldon Ross, Introduction to Probability Models, Academic Press
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