Probability Models and Methods
| Code | School | Level | Credits | Semesters |
| MATH2046 | Mathematical Sciences | 2 | 20 | Full Year Malaysia |
- Code
- MATH2046
- School
- Mathematical Sciences
- Level
- 2
- Credits
- 20
- Semesters
- Full Year Malaysia
Summary
In the first part of this module, the ideas of probability introduced in MATH1040 are extended to provide a more formal introduction to the theory of probability and random variables, with particular attention being paid to continuous random variables. Fundamental concepts, such as independence, conditioning, moments, joint distributions, transformations and generating functions are discussed in detail. This part concludes with an introduction to some probabilstic inequalities, limit theorems and the multivariate normal distribution.
The second part of the module gives an introduction to stochastic processes, i.e. random processes that evolve with time. The focus is on discrete-time Markov chains, which are fundamental to the wider study of stochastic processes. Topics covered include transition matrices, recurrence and transience, irreducibility, periodicity, equilibrium distributions, ergodic theorems, absorption probabilities, mean passage times and reversibility. Discrete-time renewal processes and branching processes are considered as applications. The module finishes with an introduction to simple one-dimensional random walks, including sample path diagrams, the reflection principle, recurrence and transience, first passage probabilities and arcsine laws. Computer Lab sessions will be used to illustrate the theory using a statistical computer package.
Target Students
Single Honours and Joint Honour Students from School of Mathematical Sciences only.
Classes
- One 1-hour tutorial each week for 24 weeks
- One 2-hour lecture each week for 24 weeks
Assessment
- 10% In-class Exam 1 ( Written): 40 min inclass test
- 10% In-class Exam 2 (Written): 40 min inclass test
- 80% Exam 1 (3-hour): 3 hours written examination
Educational Aims
The purpose of this module is to provide a thorough grounding in a broad range of techniques required in the analysis of probabilistic and statistical models, and to provide an introduction to stochastic processes by studying techniques and concepts common in the analysis of discrete time Markov Chains.Students will acquire knowledge and skills of probabilistic methods used in all subsequent modules in both Probability and Statistics.Learning Outcomes
A student who completes this module successfully will be able to:
L1 - evaluate probabilities and expectations using multivariate, marginal and conditional probability mass and density functions;
L2 - analyse transformations of one or more random variables;
L3 - derive results about and properties of random variables using generating functions;
L4 - apply results on basic limit theorems and the multivariate normal distribution in context;
L5 - state and prove basic properties of discrete-time Markov chains;
L6 - calculate standard quantities for discrete-time Markov chains (such as those concerning transition matrices, classification of states,equilibrium distributions and mean passage times);
L7 - state, derive and apply basic properties of discrete-time renewal processes, branching processes and one-dimensional random walks.
Conveners
- Dr Liew Kian Wah