# 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 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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### 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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### $\int_{G}E[X|\mathcal{G}]dP = \int_{G}XdP$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\sigma$-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 \$\...