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.

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Independence in terms of sigma fields

I was solving a homework problem that asked for an example of uncorrelated but dependent random variables. I'm using the example $X$ is uniformly distributed on $[-1,1]$, and $Y=X^2$. It's ...
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How to express a likelihood function conditioned on a filtration?

Suppose we have the sequence of discrete dependent variables $\{Y_t\}_{t=1}^n$ and we are interested to express its joint likelihood function conditioned on an entire history of a process $X$, where $...
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Determine a product's rating given a known 3-sigma tolerance

Consider a 100Ω resistor with a 10% tolerance. We can assume this is the 3-sigma value since this is typical in manufacturing. Thus, we can expect ≈99.73% of such resistors to range from 90Ω to 110Ω. ...
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How to determine sample space, $\sigma$-algebra and probability measure from the exponential family?

The sample space of binomial distribution is the set $\{0,1\}$ and its $\sigma$-algebra is the power set of $\{0,1\}$ while the sample space of normal distribution is $\mathbb R$ and its $\sigma$-...
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Why is the probability measure P not defined on the power set of $\Omega$, but on it's sigma algebra? [duplicate]

why do we not define P on the power set of $\Omega$, which is equivalent to declaring that all sets are measurable. Even if $\Omega$ is uncountable, it's sigma-algebra and power set would also be- ...
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Is there a sigma-algebra for the parameter space in relation to the likelihood function?

The likelihood function, $L(\theta|x) = f(x|\theta)$, is sometimes mistaken to be a pdf. I have always thought that showing an example where $ \int_{-\infty}^{\infty} L(\theta|x) d \theta = 1 $ does ...
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Do the following equations for the conditional expectations hold under the given assumptions?

Let $(\Omega,F,\mathbb{F}=(F_n)_{n\in\mathbb{N_0}},P)$ be a filtered probability space and $(Z_n)_{n\in\mathbb{N_0}}$ be a sequence of integrable iid random variables. Denote $F_n=\sigma(Z_o,\dots,Z_n)...
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How to understand conditional expectation w.r.t sigma-algebra: is the conditional expectation unique in this example?

This board has many questions on how to understand conditional expectations w.r.t a $\sigma$-algebra; it's clearly a topic that confuses many. I am one of them. I read all the other questions and ...
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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}(\...
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How to intuitively visualize measure theory concepts via diagrams? [closed]

Visualizing Measure Theory via Diagrams / Drawings / Geometry Hi, I'm starting to learn about Measure Theory in order to study more advanced topics in Data Science. However, the explanations are ...
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Sigma algebra generated by random variable on a set with generators

I'm having trouble proving an intuitive result I found in these lecture notes I'm using for self-study (1.2.14 there). Suppose $X$ is a $(\mathbb{S}, \mathcal{S})$-valued random variable (from $(\...
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Different modes of stating probability law [duplicate]

I met two versions of expressing induced probability measure on $\mathbb{R}^n$ of n-dimensional random vector $\mathbf{X} = (X_1, \ldots, X_n)$ (probability law, probability distribution) defined on ...
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Does product sigma algebra of n B(R) (borel) coincide with B(R^n)?

My question comes from different modes of probability law, which I met. I see two options in equality: set belonged to $B(\mathbb R^n)$ and cartesian product of B belonged to $B(\mathbb R)$.
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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 ...
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Why are p-values probabilities rather than likelihoods?

The p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct. Why is the p-value a ...
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covariance in 2 dimensional field

I want to know is the covariance equation which is shown below is how valid when we are in 2-dimensional fields (not 2-dimensional data)? $$Cov(x,y)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-y)}{N-1}$$ ...
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Independence of a Gaussian random variable and the product of another Gaussian random variable and a Bernoulli random variable

Let $X$ and $Y$ be two independent Gaussian random variables with mean $0$ and variance $σ^2_X$ and $σ^2_Y$ respectively. Let $Z$ be a random variable measurable with respect to $σ(Y)$ and suppose ...
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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 ...
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Difference between tight and uniformly tight random variables?

This wikipedia page implicitly says that “tight” and “uniformly tight” random variables refers to the same concept. I find this somewhat surprising. Are there contexts in which a distinction is made ...
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Help with $\sigma$-algebra probability theory concept [closed]

Let $\Omega = \{0,1\}^{\mathbb{N}} = \{\alpha=(\alpha_1,\alpha_2,...):\alpha_i \in \{0,1\}\}$ Fact. There exists a $\sigma$-algebra $\mathcal{F}$ such that for every $\beta = (\beta_1,...,\beta_n)\in\{...
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Understand conditional expectation w.r.t. sigma -algebra [duplicate]

When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is $$ E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
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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 ...
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Why is a random variable being a deterministic function of another random variable mean that it is in the sigma algebra of the other variables?

Specifically, I'm learning about martingales in class right now. Given random variables $T$, $X_1, X_2, \ldots, X_n$, textbook that I'm reading draws an equivalence between the statement that the ...
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Can a sigma field ever be same as an event space?

Suppose, The sample space of the gender of the upcoming baby is: $S=\{boy,girl\}$ then, $\sigma -field=\{ \{boy\}, \{girl\}, \{boy,girl\}, \emptyset\ \}$ Can the event of having the baby as twins ...
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6 votes
1 answer
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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, ...
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3 votes
1 answer
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What is the complement of an event in a sigma algebra?

Consider $\Omega=\{1,2,...,6\}$ and $F=\left\{\{1\},\{3\},\{5\}\right\}$. Let $A={1}$. If $F$ is to be a sigma algebra, then necessarily $A^c \in F$. What is $A^c$?
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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 ...
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Understanding Borel Sets in Relation to Distributions

Following up with my last question, I am self studying the book Elementary Stochastic Calculus with Finance in View. The author has lost me in some of the terseness of his explanation of distributions....
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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 $\mu$ ... ...
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2 answers
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Is there a reason other than conventions why a CDF must be defined for all real numbers

There are many cases when the sample space is not the entire set of real numbers (for instance a Bernoulli trial or sampling from an interval). On the one hand: for the definition $F_X(x) = P(X \leq ...
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$\int_{G}E[X|\mathcal{G}]dP = \int_{G}XdP$ implies that $E[X|\mathcal{G}]=X$

I need to prove that for $\mathcal{G} = \mathcal{F}$ (so the largest sigma field), $\int_{G}E[X|\mathcal{G}]dP = \int_{G}XdP\, \forall G \in \mathcal{G}$ implies that $E[X|\mathcal{F}] = X$ almost ...
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41 votes
2 answers
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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 ...
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Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$

Let $\Omega =\{1,2,3,4\}$, and let $\mathscr A = \{\{1\},\{2\}\}$. Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$. My answer is $\sigma(\mathscr A) = \mathscr A \cup \mathscr A^c =...
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2 votes
1 answer
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Computing the frequency into which x falls into the Borel $\delta$-algebra

I'm currently trying to make sense of the papers: Clustering processes by Daniil Ryabko (link) Online Clustering of Processes by Azadeh Khaleghi (link) Both of them make use of the following metric ...
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1 answer
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How to work with an asymmetrical distribution

I'm looking at estimated and actual time taken for a range of projects, as these vary in length quite a lot I've normalised them so they are just estimate/actual, so an estimate of 16 days work that ...
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4 votes
1 answer
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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{...
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3 votes
2 answers
326 views

A sigma field question

This is real basic, but a question that is annoying me: $\Omega = \left \{ 1,2,3 \right \}$ $T = \left \{ \emptyset , \Omega , \left \{ 1,2 \right \}, \left \{ 2,3 \right \}, \left \{ 1,3 \right \}...
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3 answers
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What is the importance of the concept of probability space? [duplicate]

We know sample space (S), elementary outcomes and events. And axiomatically define probability of these events or assign probabilities to these events. Now event space (F) and the triplet ...
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Distribution of X-U(0,1) conditioned on sigma algebra of Y/X, where is Y is U(0,1)?

The question I have is: Define X,Y to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$ Compute $P(X<x|\sigma(Z))$ The answer given apparently by "straightforward elementary ...
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10 votes
1 answer
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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 ...
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2 votes
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Definition of random variable with respect to event space

It is given that: We toss two dice with sample space $\Omega = \{ (i, j), 1 \leq i, j \leq 6 \}$ and the $\sigma$-algebra is generated by the events $A_k = \{ (i,j) : \max(i,j) = k \}$ $(k = 1, \...
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10 votes
1 answer
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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 ...
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31 votes
4 answers
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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 ...
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2 votes
1 answer
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Can someone give a numerical example for the definition of a field?

This is the definition for a field of events from a lecture slide: Let $\Omega$ be a set, $\mathcal{A}$ $\subset$ $P(\Omega)$ with the power set $P(\Omega)$. $\mathcal{A}$ ist called field (of ...
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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 $A_1, A_2, ... \in \mathscr{F}$...
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183 votes
4 answers
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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) ...
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Showing $X^{-1}(\mathcal F^\prime)$ is $\sigma$-algebra [closed]

Let $(\Omega,\mathcal F)$ and $(\Omega^\prime,\mathcal F^\prime)$ be two measurable spaces and $X:\Omega \to \Omega^\prime $, show that $X^{-1}(\mathcal F^\prime)$ is a $\sigma$-algebra. I know that ...
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3 votes
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Why are the Borel subsets on $\mathbb R$ a $\sigma$-algebra?

I am a newbie in mathematical statistics and haven't learnt any group theory before. My lecture notes are too brief. How can the Borel subsets on $\mathbb R$ satisfy A.2 and A.3 of a $\sigma$-algebra ?...
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6 votes
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$\sigma$-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 $\...
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