# 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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### Actual definition of $\sigma$ algebra generated by information available up to time t-1 in ARCH, GARCH models

I'm trying to figure out the proper mathematical definition in sigma-algebra generated by infinite past in ARCH, GARCH type models. Let $\epsilon_t = \sqrt{h_t} z_t$, where $z_t$ $i.i.d$ random ...
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### Calculating μ(A\ B) in probability space

I am new in learning sigma-algebra. My question is related to computing μ(A\ B). I just need the hints and ideas on how to do it, I will do it. Let (X,Ao,μ) be a probability space. It is given that A, ...
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### What is the probability that multiple spectral lines are just noise?

In spectroscopy spectral line emissions are related to each other, with fixed widths and heights that correlate with all lines based on fixed physical conditions. If I detect one spectral line at a 3 ...
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### How do we choose the sigma algebra for a specific experiment?

I am trying to understand the connect between probability and measure theory. I get some basic things about measure theory. But what I still don't get is how do we choose the sigma algebra for our ...
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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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### 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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### 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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### 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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### 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 ...