Module Catalogues, Xi'an Jiaotong-Liverpool University

Module Code: MTH302
Module Title: Applied Probability
Module Level: Level 3
Module Credits: 5.00
Semester: SEM2
Originating Department: Mathematical Sciences
Pre-requisites: MTH113MTH206

 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 occuring over time. To familiarise students with an important area of probability modelling.
 Learning outcomes A. Demonstrate a good understanding of the basic concepts related to common random processes such as Markov chains, Poisson processes, birth-death processes, Wiener process and Brownian motion.B. Give examples of Markov chains, classify the states and types for both discrete and continuous-time Markov chains.C. Define a stationary distribution for a Markov chain, be able to analyze the existence and uniqueness of a stationary distribution by using limit theorems and determine it if it exists.D. Use the properties of Poisson processes to determine the conditional distribution of arrival times.
 Method of teaching and learning
 Syllabus 1. Preliminaries: Probability generating function and its applications.2. Discrete- time Markov chains: a) Transition matrices, transition graphs Campman-Kolmogorov equations, Markov property with reference to practical examples such as simple random walks, branching processes, Bernoulli processes; b) Higher order transition probabilities, mean recurrence time; c) Classification of states/chains; d) Stationary distributions, limit theorems and asymptotic behaviour of Markov chains3. Poisson processes: a) Exponential distribution, probability of first failure, minima, sums and random sums of iid exponentials, infinitesimal behaviour, inter-occurrence times, b) Definition and properties of Poisson processes; c) Conditional distribution of the arrival times; d) Birth processes4. Continuous-time Markov chains: a) Transition rates; b) Kolmogorov forward and backward equations; c) Classification of states; d) Stationary distributions; e) Birth-death processes and examples, stationary distribution of a birth-death process5. Brownian motions and Wiener processes: a) Definition of a standard Brownian motion, i.e. Wiener process, definition of a general Brownian motion; b) Distribution of a Brownian motion 6. The following items will be introduced if time permits: Variable rate Poisson process; Compound Poisson process; Spatial Poisson process. 
Delivery Hours
 Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total Hours/Semester 39 13 98 150

## Assessment

 Sequence Method % of Final Mark 1 Final Exam 80.00 2 Class Test 1 10.00 3 Class Test 2 10.00
 Module Catalogue generated from SITS CUT-OFF: 6/5/2020 8:47:36 PM