Why do we need sigma-algebras to define probability spaces?
Mar 1, 2016 · Basically, $\sigma$ -algebras are the "patch" that lets us avoid some pathological behaviors of mathematics, namely non-measurable sets. The three requirements of a …
probability - What is it meant with the $\sigma$-algebra …
Often, in the course of my (self-)study of statistics, I've met the terminology "$\\sigma$-algebra generated by a random variable". I don't understand the definition on Wikipedia, but most …
probability - Expectation conditional on a sigma algebra, what ...
May 22, 2024 · "When we condition on the sigma algebra, do we mean that we describe all the conditional values, including conditional on the unions?" Again, this is not how rigorous …
Intuition for Conditional Expectation of $\\sigma$-algebra
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 ( …
Sigma algebra generated by random variable on a set with …
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 …
probability - How to think of a sub-sigma algebra as a collection …
Jan 28, 2023 · A random variable determines a sub sigma algebra (its events are the inverse images of the measurable sets of real numbers). Conversely, a sub sigma algebra determines …
How should I denote the smallest $\\sigma$-algebra on $\\Theta
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 …
Underlying sigma-algebra of a random process - Cross Validated
Aug 14, 2024 · Underlying sigma-algebra of a random process Ask Question Asked 1 year, 4 months ago Modified 1 year, 3 months ago
probability - Atoms of a sigma algebra - Cross Validated
Oct 18, 2018 · Atoms of a sigma algebra Ask Question Asked 7 years, 1 month ago Modified 1 year, 8 months ago
How to understand conditional expectation w.r.t sigma-algebra: is …
Nov 8, 2020 · The PMF of a discrete distribution, for example, is a Radon-Nikodym derivative wrt counting measure. On the other hand, the definition of conditional expectation requires …