# Cantor distribution

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 |

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 *C*_{t} (*t* ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in *C*_{t} containing the Cantor-distributed random variable is identically 2^{−t} on each one of the 2^{t} 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

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 *C*_{1}, 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

where *B*_{2n} is the 2*n*th Bernoulli number, and then expressing the moments as functions of the cumulants.

## References

- 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

- Morrison, Kent (1998-07-23). "Random Walks with Decreasing Steps" (PDF). Department of Mathematics, California Polytechnic State University. Retrieved 2007-02-16.