MS-E1600 - Probability Theory D, 11.01.2021-22.02.2021
Kurssiasetusten perusteella kurssi on päättynyt 22.02.2021 Etsi kursseja: MS-E1600
Osion kuvaus
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The course has 12 lectures, and each lecture corresponds to one chapter in the lecture notes:
- K Kytölä, Probability Theory, 2020
Each lecture is preceded by an online quiz which helps you to identify preliminaries from earlier courses and lectures that you may need to review independently before a lecture. Points from quizzes amount to up to 10% of the course grade, see the course syllabus for details. The deadline for completing a quiz is the start of the corresponding lecture.
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DL Mon 11 Jan 2021 at 10:00
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Introduction, sigma-algebra, generating a sigma-algebra, Borel sigma-algebras.
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Measures and probability measures.
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Measurable functions and random variables, verifying measurability, indicator functions and Dirac measures, pointwise operations on measurable real-valued functions
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Function-generated sigma-algebras, Borel preimages, Doob's representation theorem, approximating by finite-range functions
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Independent sigma-algebras, random variables, and events. Verifying independence. Borel-Cantelli lemmas.
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Tail sigma-algebra, Kolmogorov's 0-1 law
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Details about defining the natural order, topology, and Borel sigma-algebra on the extended real line.
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Theory of integration à la Lebesgue: definitions, monotonicity, linearity, approximation by sums
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Lebesgue integrals against probability measures are expectations of random variables. Lebesgue integrals against Lebesgue measures extend Riemann integrals.
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Products of sets, sigma-algebras, and measures.
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Random vectors, joint distributions, marginal distributions.
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Convergence almost surely and in probability. Weak and strong laws of large numbers.
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Fourier transforms and weak convergence of probability measures. Central limit theorem.