Cantor distribution

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Cantor

Probability mass function
Cumulative distribution function Image:CantorFunction.png Cumulative distribution function of the Cantor distribution
Parameters	none
Support	Cantor set
Probability mass function (pmf)	none
Cumulative distribution function (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
Entropy
Moment-generating function (mgf)
Characteristic function

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 a continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.

Its cumulative distribution function 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 (countably infinite) intersection of the 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.

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 B_2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

External links

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