Cantor distribution

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.

Characterization

The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:

${\displaystyle \begin{array}{cc}{C}_0=& [0,1]\\ C_1=& [0,1/3]\cup [2/3,1]\\ C_2=& [0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]\\ C_3=& [0,1/27]\cup [2/27,1/9]\cup [2/9,7/27]\cup [8/27,1/3]\cup \\ & [2/3,19/27]\cup [20/27,7/9]\cup [8/9,25/27]\cup [26/27,1]\\ C_4=& \cdots \end{array}}$

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.