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- Dictionaryindependence/ˌɪndɪˈpɛnd(ə)ns/
noun
- 1. the fact or state of being independent: "Argentina gained independence from Spain in 1816" Similar Opposite
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Sep 27, 2021 · The independence of $\mathbb R^m$-valued random variables (rv) are defined through cumulative distribution function (cdf) which is straightforward. Could you shed some light on how to generalize the definition of independence of vector-valued rv, for example, rv that values in a Banach space?
Sep 9, 2016 · $\begingroup$ Jus to add, mutual independence as seen above is a harder claim and implies pairwise indolence but it is not so the other way round. $\endgroup$ – jia chen Commented Feb 2, 2020 at 11:25
Mathematical Definition of Linear Independence. Let S be the set of vectors S = {V1, V2, V3,…..,Vn} The set S is linearly independence if and only if CV1+ C2V2 + C3V3 +….+ CnVn=zero vector The condition of checking linear independence if c1 and c2 are both zero then the two vectors are linearly independent. But Why This Formulas Make Sense?
May 2, 2017 · Roughly speaking, affine independence is like linear independence but without the restriction that the subset of lower dimension the points lie in contains the origin. So three points in space are affinely independent if the smallest flat thing containing them is a plane. They're affinely dependent if they lie on a line (or are the same point).
Feb 7, 2023 · Why do we define independence in a way that allows for an event of probability zero to be independent of another event? The case you are referring to can be seen as degenerate. If one event has probability $0$ then there isn't a clear way to interpret the notion of independence intuitively (in order to know if an event occurring has some impact on another event, we first need that event to be possible).
May 27, 2016 · Can someone define independence of two random variables with this "product rule", or are there any counterexamples?
How do we define independence for random vectors?. Eg, let $\mathbf{e}$ be a random n by 1 vector and $\mathbf{b}$ be a random k by 1 vector (where n does not necessarily have to equal to k), then how do we define independence between $\mathbf{e}$ and $\mathbf{b}$?
Jul 5, 2015 · Two events are "independent" (that is, P(E ∩ F) = P(E)P(F) P (E ∩ F) = P (E) P (F) ) if the outcome of each has no influence at all on the other. For example if we each roll a die and define E = "I roll a 6" and F = "you roll a 3". Whether E happens or not, it makes no difference to the probability of F happening.
Jun 3, 2019 · I would like your help to understand which assumptions are sufficient to get a desired conclusion about independence between random variables.
Nov 30, 2021 · Independent random variables [ edit] The theory of $\pi$-system plays an important role in the probabilistic notion of independence.
