Registry

Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Probability II
Module Code	MS208
School	School of Mathematics
Online Module Resources
Module Co-ordinator	Semester 1: Eberhard Mayerhofer Semester 2: Eberhard Mayerhofer Autumn: Eberhard Mayerhofer
NFQ level	8	Credit Rating	5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

This module aims to introduce the students to the main techniques used when dealing with several random variables. To develop the students ability to solve problems of a probabilistic nature. To motivate and explain many of the notions of convergence of sequences of random variables. To prove some of the main limit theorems. To discuss random walks. Students will attend lectures on the course material and will work, independently, to solve problems on topics related to the course material. The students will have an opportunity to review their solutions, with guidance, at weekly tutorials.

Learning Outcomes

1. Express probabilities in terms of multible integrals and be able to evaluate such integrals
2. State selected definitions and theorems
3. Solve problems that require the use of either one, several or infinitely many random variables
4. Demonstrate an understanding of concepts by use of examples or counterexamples
5. Explain arguments used to prove selected theorems

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecture	24	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	12	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent learning	89	Corresponding to each lecture students will devote approximately one and a half additional hours of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three and a half hours to solving homework problems which are to be discussed in that tutorial.
Total Workload: 125

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

SEVERAL RANDOM VARIABLES.
Joint distribution, conditional distribution, sum of random variables, distribution of a function of several random variables..

JOINT NORMAL RANDOM VARIABLES.
Definition of a Joint Normal random variabl, its covariance matrix, density function, moment generating function..

CONVERGENCE OF RANDOM SEQUENCES:.
Almost sure convergence. Convergence in L^2, in L^1, in probability, in distribution. Laws of large numbers, central limit theorem..

SIMPLE RANDOM WALK:.
Counting the number of sample paths. the reflection principle, applications..

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Oral presentations	In-class test	25%	Week 24

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List

Grimmett, G.R. and Stirzaker, D.R.:: 1992, Probability and Random Processes,, Oxford University Press,
Ross, S.: 2002, A First Course in Probability,, Prentice Hall,,

Other Resources

None

Array

Programme or List of Programmes

ACM

BSc Actuarial Mathematics

BSSA

Study Abroad (DCU Business School)

BSSAO

Study Abroad (DCU Business School)

CAFM

Common Entry into Mathematical Sciences