Cantor distribution: Difference between revisions
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| name = Cantor |
| name = Cantor |
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| 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 --> |
| 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 --> |
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| cdf_image =[[File:CantorEscalier.svg|325px|Cumulative distribution function for the Cantor distribution]]| |
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| parameters = none |
| parameters = none |
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| support = [[Cantor set]] |
| support = [[Cantor set]] |
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This distribution has neither a [[probability density function]] nor a [[probability mass function]], as it is not [[absolute continuity|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]]. |
This distribution has neither a [[probability density function]] nor a [[probability mass function]], as it is not [[absolute continuity|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]]. |
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Its cumulative distribution function is sometimes referred to as the [[Singular function|Devil's staircase]], although that term has a more general meaning. |
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. |
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== Characterization == |
== Characterization == |
Revision as of 09:00, 2 March 2016
Cumulative distribution function ![]() | |||
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 | ||
Excess kurtosis | −8/5 | ||
MGF | |||
CF |
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 continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning.
Characterization
The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets
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 2−t 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:
From this we get:
A closed-form expression for any even central moment can be found by first obtaining the even cumulants[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.