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Probability Distributions01:32

Probability Distributions

The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
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Distribution and Dispersion00:54

Distribution and Dispersion

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Chi-square Distribution01:10

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Related Experiment Video

Updated: Jul 19, 2026

Using Wavelet Entropy to Demonstrate how Mindfulness Practice Increases Coordination between Irregular Cerebral and Cardiac Activities
08:08

Using Wavelet Entropy to Demonstrate how Mindfulness Practice Increases Coordination between Irregular Cerebral and Cardiac Activities

Published on: May 10, 2017

Relations between distributional and Devaney chaos.

Piotr Oprocha1

  • 1Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 39, 30-059 Kraków, Poland. opracha@agh.edu.pl

Chaos (Woodbury, N.Y.)
|October 4, 2006
PubMed
Summary

Distributional chaos does not imply Li-Yorke chaos, contrary to previous findings for Devaney chaos and weak mixing. This study provides counterexamples demonstrating the distinction between these chaos definitions.

Area of Science:

  • Dynamical Systems
  • Chaos Theory

Background:

  • Recent work established that Devaney chaos and weak mixing imply Li-Yorke chaos.
  • However, the relationship between these definitions and distributional chaos remained unclear.

Purpose of the Study:

  • To investigate whether Devaney chaos or weak mixing imply distributional chaos.
  • To determine if distributional chaos implies Li-Yorke chaos.

Main Methods:

  • Construction of explicit mathematical examples.
  • Analysis of dynamical systems exhibiting specific chaotic behaviors.

Main Results:

  • Demonstrated that Devaney chaos does not imply distributional chaos.
  • Showcased that weak mixing does not imply distributional chaos.

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08:08

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Conclusions:

  • The implications from Devaney chaos and weak mixing to Li-Yorke chaos do not extend to distributional chaos.
  • Distributional chaos is a distinct concept from other established notions of chaos.