Module Specifications..
Current Academic Year 2023  2024
Please note that this information is subject to change.
 
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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  
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, BorelCantelli lemma, Levy's continuity theorem. Limit theorems: Weak and strong laws of large numbers, central limit theorem, BerryEsseen's theorem.  
 
Indicative Reading List
 
Other Resources None  
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