World Library  
Flag as Inappropriate
Email this Article

Cantor distribution

Article Id: WHEBN0000387941
Reproduction Date:

Title: Cantor distribution  
Author: World Heritage Encyclopedia
Language: English
Subject: Probability distribution, Singular distribution, Variance, Singular measure, Probability distributions
Collection: Georg Cantor, Probability Distributions
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Cantor distribution

Cantor
Parameters none
Support Cantor set
pmf none
CDF Cantor function
Mean 1/2
Median anywhere in [1/3, 2/3]
Mode n/a
Variance 1/8
Skewness 0
Ex. kurtosis −8/5
MGF e^{t/2} \prod_{i= 1}^{\infty} \cosh{\left(\frac{t}{3^{i}} \right)}
CF e^{\mathrm{i}\,t/2} \prod_{i= 1}^{\infty} \cos{\left(\frac{t}{3^{i}} \right)}

The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.

This distribution has neither a probability density function nor a probability mass function, as it is not absolutely continuous with respect to Lebesgue measure, nor has it any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.

Its cumulative distribution function is sometimes referred to as the Devil's staircase, although that term has a more general meaning.

Contents

  • Characterization 1
  • Moments 2
  • References 3
  • External links 4

Characterization

The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets

\begin{align} C_{0} = & [0,1] \\ C_{1} = & [0,1/3]\cup[2/3,1] \\ C_{2} = & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\ C_{3} = & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\ & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\ C_{4} = & \cdots . \end{align}

The Cantor distribution is the unique probability distribution for which for any Ct (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in Ct containing the Cantor-distributed random variable is identically 2t on each one of the 2t intervals.

Moments

It is easy to see by symmetry that for a random variable X having this distribution, its expected value E(X) = 1/2, and that all odd central moments of X are 0.

The law of total variance can be used to find the variance var(X), as follows. For the above set C1, let Y = 0 if X ∈ [0,1/3], and 1 if X ∈ [2/3,1]. Then:

\begin{align} \operatorname{var}(X) & = \operatorname{E}(\operatorname{var}(X\mid Y)) + \operatorname{var}(\operatorname{E}(X\mid Y)) \\ & = \frac{1}{9}\operatorname{var}(X) + \operatorname{var} \left\{ \begin{matrix} 1/6 & \mbox{with probability}\ 1/2 \\ 5/6 & \mbox{with probability}\ 1/2 \end{matrix} \right\} \\ & = \frac{1}{9}\operatorname{var}(X) + \frac{1}{9} \end{align}

From this we get:

\operatorname{var}(X)=\frac{1}{8}.

A closed-form expression for any even central moment can be found by first obtaining the even cumulants[1]

\kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}} {n\, (3^{2n}-1)}, \,\!

where B2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

References

External links

  • Morrison, Kent (1998-07-23). "Random Walks with Decreasing Steps" (PDF). Department of Mathematics, California Polytechnic State University. Retrieved 2007-02-16. 
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.