2025年2月18日 · The meaning of Sigma in statistical analysis is a function of whether you use the upper or lower case of the letter. The uppercase use of sigma denotes a summation while the lowercase represents a measure of variation. Overview: How Is Sigma Used in Statistical Analysis? Below are the uppercase and lowercase symbols for sigma:

2016年3月27日 · Is there a math function, similar to sigma, that can run down? For example instead of $\sum\limits_{i=1}^{10}i$ , something that adds from 10 to 1 (like a backwards run)...

2015年1月23日 · the essential property of sigma-additivity that is used to define sigma-algebra was first used both by Émile Borel in 1898 for his work on the theory of functions in which there is a chapter on the measure theory:

2013年10月16日 · The letters $\sigma$ and $\delta$ are often given as Greek abbreviations of German words: $\sigma$ as S in Summe for sum (in the sense of sum of sets, that is, union) and $\delta$ as D in Durchschnitt for intersection, both countable. Thus, in the context of measure theory, the letter $\sigma$ refers to the stability of a collection of subsets ...

2019年3月21日 · It's not a matter of "notation": $\Pi$ is for products and $\Sigma$ is for sums. If each factor in the product is a sum of a few terms, you are expanding out the product: each choice of one of the terms in each factor gives you a term of …

In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate

As a corollary, there is no countably infinite sigma algebra. See, for example, "A comment on unions of sigma fields, A. Broughton, B. Huff, American mathematical monthly, 1977 Vol. 84 No. 7, pp 553-54".

$\begingroup$ The basic two trivial $\sigma$-algebra definition I got was, (empty set and the whole set) due to closed under complementation, and all possible subsets due to closed under union. But what would be a bit non trivial example which would explain the interplay between the closure under complementation and union. $\endgroup$

How can one calculate the Pinsker $\sigma$-algebra of the Bernoulli shift $\left(\dfrac{1}{2},\dfrac{1}{2}\right)$? I think that the Pinsker $\sigma$-algebra is the $\sigma$-algebra of all measurable sets of measure $0$ or $1$. And another question: Why is the Pinsker $\sigma$-algebra important in ergodic theory?

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