It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these.

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 2−t 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.

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.