Module: Probability II

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Probability II
Module Code	MTH1044 (ITS: MS232)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	8	Credit Rating	7.5

Description

This module aims to introduce the students to the main techniques used when dealing with several random variables. It also offers an introduction to the limit theory of sequences of random variables. 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

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
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 Study	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

Section Breakdown
CRN	20788	Part of Term	Semester 2
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC3	Best Mark	Y
Module Co-ordinator	Martin Venker	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
In Class Test	n/a	20%	As required
Formal Examination	End-of-Semester Final Examination	80%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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

RANDOM VECTORS:
Joint distribution, marginal distributions, conditioning on a random variable, transformation of random variables, independent random variables, convolution, characteristic function.

JOINTLY NORMAL RANDOM VARIABLES:
Equivalent characterisations of jointly normal random variables using mean vector and covariance matrix, densities, characteristic function or linear marginals. Linear transformations and conditioning.

LIMIT THEORY FOR SEQUENCES OF RANDOM VARIABLES:
Modes of convergence: almost sure convergence, convergence in L^p, in probability, in distribution. Relations between different modes of convergence. Tools for proving convergence: Chebyshev's and Markov's inequalities, Borel-Cantelli lemma, Levy's continuity theorem. Limit theorems: Weak and strong laws of large numbers, central limit theorem, Berry-Esseen's theorem.

Indicative Reading List

Books:

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,

Articles:
None

Other Resources

None