So for an event,A, the sigma algebra is $ \ {A,A^C,\emptyset, \Omega\}$, so clearly it is set as you specified in your question for $\omega \in A/A^c$. Similarly for any discrete random variable ( …

Often, in the course of my (self-)study of statistics, I've met the terminology " σ σ -algebra generated by a random variable". I don't understand the definition on Wikipedia, but most …

Jan 2, 2014 · So, the order of the operations of "taking inverse images" and "generating a $\sigma$ -algebra" doesn't matter. Is this understanding correct? Any hint on a direction that …

May 22, 2024 · Expectation conditional on a sigma algebra, what expectation does it refer to? Ask Question Asked 1 year, 7 months ago Modified 1 year, 7 months ago

Aug 14, 2024 · Underlying sigma-algebra of a random process Ask Question Asked 1 year, 4 months ago Modified 1 year, 3 months ago

Feb 8, 2025 · 2 Aim for simplicity/clarity Given that $\Sigma$ is the sigma-algebra you started with, and you are now forming a minimal sigma-algebra from this, I would recommend using …

Aug 14, 2024 · Sigma-algebras (or $\sigma$-fields or $\sigma$-algebras) define which subsets of the sample space $\Omega$ are considered events for the purposes of computing …

Oct 18, 2018 · Hence all Ai A i are atoms of the given sigma algebra. To see why no strict subset can be constructed, draw a diagram of a space and partition it into disjoint parts.

Oct 17, 2023 · Consider this "less advanced" proof: Let , where are independent random variables and are measurable functions. Then: Using independence of and , The idea is to notice that …

Aug 20, 2020 · The sigma field for y y is generated by the inverse images of any pi-system that generates the Borel sets of R. R . A standard pi system consists of the sets of the form (−∞, a] …