Essentials in Analysis and Probability (MATH10047)
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The central topic of this course is measure theory. Measure theory is the foundation for advanced topics in Analysis and Probability.
The course will cover many of the following topics: Random events, sigma-algebras, monotone classes.Measurable spaces, random variables - measurable functions.Measures, probability measures, signed measures.Borel sets in R^d, Lebesgue measure. Caratheodory extension theorem. Sequences of events and random variables, Borel-Cantelli lemma.Distributions of random variables. Independence of random variables.Integral of measurable functions - mathematical expectation,.Moments of random variables, L_p spaces.Convergence concepts of measurable functions.Limit theorems for integrals.Weak and strong laws of large numbers.Completeness of L_p spaces.Conditional expectation and conditional distribution of random variables.Fubini's theorem.
Written Exam 95%, Coursework 5%, Practical Exam 0%
Additional Assessment Information
Coursework 5%, Examination 95%There will be 5 assignments. Each assignment will be marked out of 20; a mark of 7 or lower on an assignment will be recorded as no credit, and a mark of 8 or higher will be recorded as full credit. Some assignment questions are harder than others. The assignment will be due by W3, W5, W7 W9 and W11. At the end of the semester, the best 4 out of 5 assignments will be counted.
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