Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [sigma-algebra]

Sigma-algebras (or $\sigma$-fields or $\sigma$-algebras) define which subsets of the sample space $\Omega$ are considered events for the purposes of computing probabilities. Sigma-algebras are often denoted $\mathscr{F}$. They are used in the definition of a probability space which is a triple $(\Omega, \mathscr{F}, \mathbb{P})$ which precisely defines how probabilities will be computed.

Sigma-algebras (or $\sigma$-fields or $\sigma$-algebras) define which subsets of the sample space $\Omega$ are considered events for the purposes of computing probabilities. Sigma-algebras are often denoted $\mathscr{F}$. They are used in the definition of a probability space which is a triple $(\Omega, \mathscr{F}, \mathbb{P})$ which precisely defines how probabilities will be computed.

A $\sigma$-field is a set that has three properties:

  1. Closure under countable unions.
  2. Closure under countable intersections.
  3. Closure under complements.

We are considering events in relation to $\Omega$, so we further require that $\Omega\in\mathscr{F}$

49 questions
182
votes
4 answers

Why do we need sigma-algebras to define probability spaces?

We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) are made up of events to which a probability…
Antoni Parellada
  • 23,430
  • 15
  • 100
  • 197
40
votes
2 answers

What is it meant with the $\sigma$-algebra generated by a random variable?

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 importantly I don't get the intuition behind it. Why/when do…
DeltaIV
  • 15,894
  • 4
  • 62
  • 104
31
votes
4 answers

Intuition for Conditional Expectation of $\sigma$-algebra

Let $(\Omega,\mathscr{F},\mu)$ be a probability space, given a random variable $\xi:\Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$ we can construct a new random variable $E[\xi|\mathscr{G}]$, which is the…
Nicolas Bourbaki
  • 1,080
  • 9
  • 22
10
votes
1 answer

Computation of Conditional Expectation on $\sigma$-algebras

I have not really seen any probability books calculate conditional expectation, except for $\sigma$-algebras generated by a discrete random variable. They simply state the existence of conditional expectation, along with its properties, and leave it…
Nicolas Bourbaki
  • 1,080
  • 9
  • 22
8
votes
1 answer

Definition of sample space

From Rohatgi-Saleh's book on probability and statistics: Def: The sample space of a statistical experiment is a pair $(\Omega,\mathcal S)$, where (a) $\Omega$ is the set of all possible outcomes of the experiment. (b) $\mathcal S$ is a…
StubbornAtom
  • 8,662
  • 1
  • 21
  • 67
7
votes
1 answer

Simple example of the $\sigma$-field generated by a random variable (Concept check)

$\Omega = \{ {\omega_1, \omega_2,\omega_3} \}$ where each state is equally probable. Two random variables exist $\widetilde{x}$ and $\widetilde{y}$ that are functions of these states: $\widetilde{x}(\omega_i)=a_i$ where $a_1 \neq a_2 \neq…
financial_physician
  • 555
  • 2
  • 9
6
votes
1 answer

$\sigma$-algebra intersection of infinite subsets

I am working out a book on Lebesgue measure by Bartle, and would like to see the steps that go into the construction of a proof for the following: Show that any $\sigma$-algebra of subsets of $\mathbb{R}$ which contains all open intervals also…
hearse
  • 2,355
  • 1
  • 17
  • 30
5
votes
2 answers

Do likelihood functions require sigma-algebras as probability spaces do?

In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a function of the parameters given the data…
user271077
5
votes
1 answer

Why must probability fields be closed under countable unions?

Assume a probability triplet $(\Omega, \mathcal{F}, \mathbb{P})$. My current understanding of $\mathcal{F}$ is that it must define events i.e. the subsets of $\Omega$ where probability is defined. I also understand that $\mathcal{F}$ must be closed…
Vykta Wakandigara
  • 423
  • 3
  • 8
5
votes
1 answer

What does the meet of two sigma algebras mean?

I came across this notation of which I am unfamiliar; $\mathscr{F}=\mathscr{G}_{1}\vee \mathscr{G}_{1}$ where $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$ are both sigma-fields of subsets of $\Omega$. It is claimed $\mathscr{F}$ is larger than both of…
dandar
  • 638
  • 5
  • 14
5
votes
1 answer

Atoms of a sigma algebra

I've been reading Schilling's Measures, Integrals, and Martingales, and I ran across a remark (on page 21) that I don't understand. Here is the setup: let $A_{1},\ldots,A_{n}$ be non-empty, disjoint subsets of X with $\cup A_{n}=X$. Schilling…
Adam Bendorf
  • 53
  • 5
5
votes
1 answer

Is an event a subspace of the sample space?

In a lecture today, a professor of mine described an event as being "in" the sample space. When writing on the board, for a sample space $S$ and event $E$, it was denoted: $$E \in S $$ This confused me, as I have always thought that events were…
James Otto
  • 574
  • 5
  • 6
4
votes
0 answers

A possible typo in the textbook?

On page 74 of Lehmann's Testing Statistical Hypothesis, the author writes Let $P_0$ and $P_1$ be probability distributions possessing densities $p_0$ and $p_1$ respectively w.r.t. a measure $\mu$ ... Let $\alpha\left(c\right) = P_0 \left\{…
Yuki Kawabata
  • 221
  • 2
  • 4
4
votes
1 answer

Exact meaning of conditional expectation $\mathbb{E}[X|\mathcal{F}]$

I'm going through elementary literature on measure theory from Shreve (Vol II) and having a hard interpreting the meaning of $\mathbb{E}[X|\mathcal{F(t)}]$ where $X$ is a random variable and $\mathcal{F(t)}$ is the sigma-algebra at time $t$. Usually…
Akshay Bansal
  • 203
  • 1
  • 8
4
votes
2 answers

Is the union of all elements in a $\sigma$-field equal to $\Omega$?

From the textbook I'm reading, A collection of subsets $\mathscr{F}$ of $\Omega$ is called a $\sigma$-field if it satisfies: empty set in $\mathscr{F}$ if $A_1, A_2, ... \in \mathscr{F}$, then union of $A$'s exist in $\mathscr{F}$ if $A \in…
pigate
  • 43
  • 3
1
2 3 4