# 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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### Expectation conditional on a sigma algebra, what expectation does it refer to?

In a question like Intuition for Conditional Expectation of $\sigma$-algebra a concept like $E[X|\sigma(Y)]$ is used and I am puzzled about what sort of variable this actually is. Say we have a ...
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### Confusion about the notation $X\in\mathcal{F}_o$, where $X$ is a random variable and $\mathcal{F}_o$ is a sigma-algebra

In Durrett's Probability:Theory and Examples page 205 section 4.1, it has the following notation $X\in \mathcal{F}_o$ (see the picture below). I'm confused about this notation as $X$ is a random ...
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### Example of a set of possible outcomes which is not an event

On slide number 6 in this link, I found the following lines: Not every set of possible outcomes will be called an event. Instead, we will require that the collection of events is a $\sigma$-algebra. ...
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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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### 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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### 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 ...
### 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 ...