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 C<sub>t</sub> (t â { 0, 1, 2, 3, ... }), the probability of a particular interval in C<sub>t</sub> containing the Cantor-distributed random variable is identically 2<sup>âÂÂt</sup> on each one of the 2<sup>t</sup> 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 C<sub>1</sub>, let Y = 0 if X â [0,1/3], and 1 if X â [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<sub>2n</sub> is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.