Probability distributions | Metric geometry
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: * For all u and v in S, u = v if and only if Fu,v(x) = 1 for all x > 0. * For all u and v in S, Fu,v = Fv,u. * For all u, v and w in S, Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 for x, y > 0. (Wikipedia).
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
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From playlist Topology
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From playlist Topology
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From playlist Mathematical analysis and applications
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I started with the definition of a metric space, we briefly discussed the example of Euclidean space (proofs next time) and then I started to explain a few natural metrics on the circle. Lecture notes: http://therisingsea.org/notes/mast30026/lecture2.pdf The class webpage: http://therisin
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Visit https://brilliant.org/TreforBazett/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor Weird, fun
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This video is about metric spaces and some of their basic properties.
From playlist Basics: Topology
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From playlist MAST30026 Metric and Hilbert spaces
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Mokshay Madiman : Minicourse on information-theoretic geometry of metric measure
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From playlist Plenary talks One World Symposium 2020
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From playlist Machine Learning Tutorials