The Cantor distribution is the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

whose

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

is the Cantor function. This distribution has neither a probability density function nor a

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, since although its cumulative distribution function is a

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

, the distribution is not absolutely continuous with respect to

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

, 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 In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero. Other names These distributions are sometimes called singular continu ...

. 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

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of the Cantor distribution is the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...

, itself the intersection of the (countably infinitely many) sets: :

C_5 = & \cdots \end

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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

''X'' having this distribution, its

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

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''₁, let ''Y'' = 0 if ''X'' ∈ ,1/3 and 1 if ''X'' ∈ /3,1 Then: :

\begin
\operatorname(X) & = \operatorname(\operatorname(X\mid Y)) + 
                          \operatorname(\operatorname(X\mid Y)) \\
                      & = \frac\operatorname(X) + 
                          \operatorname
                            \left\ \\
                      & = \frac\operatorname(X) + \frac
\end

From this we get: :

\operatorname(X)=\frac.

A closed-form expression for any even central moment can be found by first obtaining the even cumulants :

\kappa_ = \frac
                    , \,\!

where ''B''_2''n'' is the 2''n''th Bernoulli number, and then expressing the moments as functions of the cumulants.

Characterization

Moments

References

Further reading