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Cantor
Cumulative distribution function
Parameters
none
Support
Cantor set, a subset of [0,1]
PMF
none
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
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 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.
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