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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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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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 ...
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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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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 ...
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 ...