The Cantor distribution is the
probability distribution whose
cumulative distribution function is the
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. ...
.
This distribution has neither a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
nor a
probability mass function, since although its cumulative distribution function is a
continuous function, the distribution is not
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
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:
:
The Cantor distribution is the unique probability distribution for which for any ''C''
''t'' (''t'' ∈ ), 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 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 In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
can be used to find the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
var(''X''), as follows. For the above set ''C''
1, let ''Y'' = 0 if ''X'' ∈
,1/3 and 1 if ''X'' ∈
/3,1 Then:
:
From this we get:
:
A closed-form expression for any even
central moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
can be found by first obtaining the even
cumulants
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
:
where ''B''
2''n'' is the 2''n''th
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, and then
expressing the moments as functions of the cumulants.
References
Further reading
* ''This, as with other standard texts, has the Cantor function and its one sided derivates.''
* ''This is more modern than the other texts in this reference list.''
*
* ''This has more advanced material on fractals.''
{{Clear
Continuous distributions
Georg Cantor