MS-E1600 - Probability Theory D, 11.01.2021-22.02.2021
This course space end date is set to 22.02.2021 Search Courses: MS-E1600
Lecture 8: Integrations and expectations
8.1 Signed integrals
Lebesgue integral of a signed function is defined by separately integrating its positive and negative parts.
8.2 Where is almost everywhere?
Values outside a set of measure zero determine the integral of a function.
8.3 Lebesgue vs Riemann
Lebesgue integral against a Lebesgue measure coincides with the classical Riemann integral for Riemann-integrable Borel functions.
8.4 Expectations
Lebesgue integrals against a probability measure are expectations of random variables.
8.5 Integrating against the law
The expectation of a random variable can be computed by integrating against the law of the random variable.
8.6 Integrating against a density
Laws of some random variables admit a probability density function with respect to the Lebesgue measure on the real line. Expectations related to such random variables reduce to familiar formulas (e.g. normal distribution, exponential distribution).
8.7 Monotone and dominated continuity
Integration as a linear functional is not continuous in general, but continuity holds for monotone and suitably dominated function sequences.
Alternative reading material
- [Jacod & Protter, Chapter 9]
- [Williams, Chapters A5, 5.4, 5.9, 6.1, 6.2, 6.5]