| CODE | SOR2120 | ||||||||
| TITLE | Convergence and Limits in Probability | ||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Statistics and Operations Research | ||||||||
| DESCRIPTION | - Modes of Convergence; - Borel-Cantelli Lemmas; - Weak Law of Large Numbers; - Strong Law of Large Numbers; - De Moivre-Laplace Theorem; - Central Limit Theorem - The Lindeberg, Feller, Lyapunov Conditions; - Weak Convergence of Measures. Study-Unit Aims: - Familiarize with the Borel-Cantelli lemma and how it can be used in a variety of different contexts; - Familiarize with the modes of convergence: almost sure convergence, convergence in probability, convergence in distribution; - Use the Borel-Cantelli lemma to prove some of the modes of convergence; - Link the modes of convergence: how almost sure convergence implies convergence in probability which in turn implies convergence in distribution; - Familiarize with the weak and strong laws of large numbers; - Be able to use the Lindeberg-Feller condition as well as the Lyapunov condition to prove the central limit theorem. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Understand the difference between the different modes of convergence and the link between them; - Familiarize with the Borel-Cantelli lemma and its use to prove some of the modes of convergence; - Familiarize with weak and strong laws of large numbers; - Use the Lindeberg-Feller and the Lyapunov condition to prove the central limit theorem. 2. Skills: By the end of the study-unit the student will be able to: - Prove mathematically the modes of converge of sequences of random variables; - Use the Borel-Cantelli Lemma to prove some of the modes of convergence; - Prove mathematically that the sample average of a sequence of random variables converges to the true population average (strong and weak law of large numbers). - Use the Lindeberg-Feller and the Lyapunov condition to prove the central limit theorem. Main Text/s and any supplementary readings: Suggested Texts: - Rohatgi V.K. ( 1975 ) An Introduction to Probability Theory and Mathematical Statistics, Wiley Series - Chung, K. L., ( 1979 ) Elementary Probability Theory with Stochastic Processes, Springer - Ross S., ( 1997 ) Introduction to Probability Models, Academic - Feller, W. ( 1971 ) An Introduction to Probability Theory and its Applications, John Wiley & Sons - Grimmett, G.R. and Stirzaker, ( 1994 ) D.R. Probability and Random Processes, Clarendon Press, Oxford. - Billingsley P. ( 1995 ) Probability and Measure, John Wiley & Sons - Bauer, Heinz ( 1981 ) Probability Theory and Elements of Measure Theory, Academic Press - McCabe B. and Tremayne A. ( 1991 ) Elements of Asymptotic Theory with Statistical Application, Manchester University Press |
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| ADDITIONAL NOTES | Pre-requisite Study-unit: SOR1110 | ||||||||
| STUDY-UNIT TYPE | Lecture | ||||||||
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| LECTURER/S | Mark A. Caruana |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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