Editing Cantor distribution

{{short description|Probability distribution}}
{{Refimprove|date=January 2017}}
{{Probability distribution
| name       = Cantor
| type       = mass <!-- not technically correct; added only so template pmf entry would parse correctly; used "mass" rather than "density" since cdf plot suggests it *might* have a pmf -->
| cdf_image  =[[File:CantorEscalier-2.svg|325px|Cumulative distribution function for the Cantor distribution]]|
| parameters = none
| support    = [[Cantor set]], a subset of [0,1]
| pdf        = none
| cdf        = [[Cantor function]]
| mean       = 1/2
| median     = anywhere in [1/3,&nbsp;2/3]
| mode       = n/a
| variance   = 1/8
| skewness   = 0
| kurtosis   = −8/5
| entropy    =
| mgf        = <math>e^{t/2} \prod_{k=1}^\infty \cosh\left(\frac{t}{3^k}\right)</math>
| char       = <math>e^{it/2} \prod_{k=1}^\infty \cos\left(\frac{t}{3^k}\right)</math>
}}

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]], since although its cumulative distribution function is a [[continuous function]], the distribution is not [[absolute continuity|absolutely continuous]] with respect to [[Lebesgue measure]], nor does it have 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 continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the [[Singular function|Devil's staircase]], although that term has a more general meaning.

== Characterization ==

The [[Support (mathematics)|support]] of the Cantor distribution is the [[Cantor set]], itself the intersection of the (countably infinitely many) sets:
: <math>
\begin{align}
 C_0 = {} & [0,1] \\[8pt]
 C_1 = {} & [0,1/3]\cup[2/3,1] \\[8pt]
 C_2 = {} & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\[8pt]
 C_3 = {} & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\[4pt]
         {} & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\[8pt]
 C_4 = {} & [0,1/81]\cup[2/81,1/27]\cup[2/27,7/81]\cup[8/81,1/9]\cup[2/9,19/81]\cup[20/81,7/27]\cup \\[4pt]
         & [8/27,25/81]\cup[26/81,1/3]\cup[2/3,55/81]\cup[56/81,19/27]\cup[20/27,61/81]\cup \\[4pt]
         & [62/81,21/27]\cup[8/9,73/81]\cup[74/81,25/27]\cup[26/27,79/81]\cup[80/81,1] \\[8pt]
 C_5 = {} & \cdots
\end{align}
</math>

The Cantor distribution is the unique probability distribution for which for any ''C''<sub>''t''</sub> (''t''&nbsp;∈&nbsp;{&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...&nbsp;}), the probability of a particular interval in ''C''<sub>''t''</sub> containing the Cantor-distributed random variable is identically 2<sup>−''t''</sup> on each one of the 2<sup>''t''</sup> intervals.

== Moments ==
It is easy to see by symmetry and being bounded 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&nbsp;0.

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

: <math>
\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}
</math>

From this we get:

:<math>\operatorname{var}(X)=\frac{1}{8}.</math>

A closed-form expression for any even [[central moment]] can be found by first obtaining the even [[cumulants]]<ref>{{cite web |last=Morrison |first=Kent |url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |title=Random Walks with Decreasing Steps |publisher=Department of Mathematics, California Polytechnic State University |date=1998-07-23 |access-date=2007-02-16 |archive-date=2015-12-02 |archive-url=https://web.archive.org/web/20151202055102/http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |url-status=dead }}</ref>

:<math>
 \kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}}
                    {n\, (3^{2n}-1)}, \,\!
</math>

where ''B''<sub>2''n''</sub> is the 2''n''th [[Bernoulli number]], and then [[Cumulant#Cumulants and moments|expressing the moments as functions of the cumulants]].

== References ==
{{Reflist}}

==Further reading==
* {{cite book |first1=E. |last1=Hewitt |first2=K. |last2=Stromberg |title=Real and Abstract Analysis |url=https://archive.org/details/realabstractanal00hewi_0 |url-access=registration |publisher=Springer-Verlag |location=Berlin-Heidelberg-New York |year=1965}} ''This, as with other standard texts, has the Cantor function and its one sided derivates.''
* {{cite news |first1=Tian-You |last1=Hu |first2=Ka Sing |last2=Lau |title=Fourier Asymptotics of Cantor Type Measures at Infinity |journal=Proc. AMS |volume=130 |number=9 |year=2002 |pages=2711–2717}} ''This is more modern than the other texts in this reference list.''
* {{cite book |first=O. |last=Knill |title=Probability Theory & Stochastic Processes |publisher=Overseas Press |location=India |year=2006}}
* {{cite book |first=P. |last=Mattilla |title=Geometry of Sets in Euclidean Spaces |publisher=Cambridge University Press |location=San Francisco |year=1995}} ''This has more advanced material on fractals.''

{{ProbDistributions|miscellaneous}}
{{Clear}}

[[Category:Continuous distributions]]
[[Category:Georg Cantor]]