Cantor distribution

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Cumulative distribution function
Cumulative distribution function for the Cantor distribution
Parameters none
Support Cantor set, a subset of [0,1]
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

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 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 Devil's staircase, although that term has a more general meaning.


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.


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 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.


  1. ^ Morrison, Kent (1998-07-23). "Random Walks with Decreasing Steps" (PDF). Department of Mathematics, California Polytechnic State University. Archived from the original (PDF) on 2015-12-02. Retrieved 2007-02-16.

Further reading[edit]

  • Hewitt, E.; Stromberg, K. (1965). Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer-Verlag. This, as with other standard texts, has the Cantor function and its one sided derivates.
  • Hu, Tian-You; Lau, Ka Sing (2002). "Fourier Asymptotics of Cantor Type Measures at Infinity". Proc. AMS. Vol. 130, no. 9. pp. 2711–2717. This is more modern than the other texts in this reference list.
  • Knill, O. (2006). Probability Theory & Stochastic Processes. India: Overseas Press.
  • Mattilla, P. (1995). Geometry of Sets in Euclidean Spaces. San Francisco: Cambridge University Press. This has more advanced material on fractals.