Cantor distribution

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Cantor
Cumulative distribution function
Cumulative distribution function for the Cantor distribution
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
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[edit]

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 2t on each one of the 2t intervals.

Recently discovered, the Geometric Mean of all reals in the Cantor Set between (0,1] is approximately 0.274974, which is ≈ 75% of the Geometric Mean of all reals in between (0,1].[1]

Moments[edit]

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 except for the first moment 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[edit]

  • Falconer, K. J. (1985). Geometry of Fractal Sets. Cambridge & New York: Cambridge Univ Press. 
  • Hewitt, E.; Stromberg, K. (1965). Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer-Verlag. 
  • Hu, Tian-You; Lau, Ka Sing (2002). "Fourier Asymptotics of Cantor Type Measures at Infinity". Proc. A.M.S. 130 (9). pp. 2711–2717. 
  • Knill, O. (2006). Probability Theory & Stochastic Processes. India: Overseas Press. 
  • Mandelbrot, B. (1982). The Fractal Geometry of Nature. San Francisco, CA: WH Freeman & Co. 
  • Mattilla, P. (1995). Geometry of Sets in Euclidean Spaces. San Francisco: Cambridge University Press. 
  • Saks, Stanislaw (1933). Theory of the Integral. Warsaw: PAN.  (Reprinted by Dover Publications, Mineola, NY.

External links[edit]