Editing Cantor distribution (section)

{{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.