Cantor distribution
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Cumulative distribution function
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Parameters | none |
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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 | |
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, since although it is a continuous function 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
[1]KJ Falconer, GEometry of Fractal Sets, Cambridge Univ Press, Cambridge & New York, 1985 and later eds. (This has a chapter on Self-Similar Fractals; does not have the Cantor Function , graphed in this article))
[2]E.Hewitt & K Stromberg, Real and abstract analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1965 and later eds( This as withother standard texts has the Cantor Function and its one sided derivates)
[3])Tian-You Hu and Ka Sing Lau, Fourier Asymptotics of Cantor Type Measures atInfinity, Proc. A.M.S.,vol 130,Number9(2002) pp2711-2717( is related to the this article and is more modern than the Texts in this Reference list)
[4]O.Knill;Probability Theory&Stochastic Processes...Overseas Press (India) 2006 ( also directly related to this article)
[5]B.Mandelbrojt, The Fractal Geometry of Nature, WH Freeman &CO;San FRancisco CA (1982) ( This is where thepopular name"Fractals" was coined; According to a Reviewer for the AMS," written with daring originality".contains much valuable material. Most of it is by now rewritten in a style suitable for/ acceptable to pure mathematicians.; some of it by Falconer and in Mattillas books listed)
[6]P.Mattilla; Geometry of sets in Euclidean Spaces, CambridgeUniv Press , San FRancisco , 1995 and later edns..( This has more advanced material on Fractals)
[7]Stanislaw Saks, Theory of the Integral; PAN, Warsaw 1933 and later eds; Reprinted by Dover Publications, Mineola, NY(Classic Gem; contains a small section on Hausdorff Measures and Hausdorff dimension= Fractal Dimension)
External links
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