The Cantor distribution is the

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

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. Ever ...

is the Cantor function. This distribution has neither a

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

nor a

probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...

, since although its cumulative distribution function is a

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

, the distribution is not

absolutely continuous In calculus and real analysis, 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 betwe ...

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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

, 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: :

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 Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

''X'' having this distribution, its

expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...

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 In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...

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 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 ...

\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