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The Cantor distribution is the probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 whose cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 is the Cantor function
Cantor function

In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous function, but not absolutely continuous....
.

This distribution has neither a probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 nor a probability mass function
Probability mass function

In probability theory, a probability mass function is a function that gives the probability that a discrete random variable random variable is exactly equal to some value....
, as it is not absolutely continuous
Absolute continuity

In mathematics, absolute continuity is a smoothness property which is stricter than continuity and uniform continuity. Both absolute continuity of functions and absolute continuity of measures are defined....
 with respect to Lebesgue measure
Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
, nor has it any point-masses. It is thus neither a discrete nor a continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution
Singular distribution

In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero set where the probability of each point in that set is zero....
.

Its cumulative distribution function is sometimes referred to as the Devil's staircase
Devil's staircase

In mathematics, a singular function is any function ƒ defined on the interval [a, b] that has the following properties:*ƒ is continuous function on [a, b]....
, although that term has a more general meaning.
Moments It is easy to see by symmetry that for a random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
 X having this distribution, its expected value
Expected value

In probability theory and statistics, the expected value of a random variable is the Lebesgue integral of the random variable with respect to its probability measure....
 E(X) = 1/2, and that all odd central moments of X are 0.

The law of total variance
Law of total variance

In probability theory, the law of total variance or variance decomposition formula states that if X and Y are random variables on the same probability space, and the variance of X is finite, then...
 can be used to find the variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
 var(X), as follows.






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Encyclopedia


The Cantor distribution is the probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 whose cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 is the Cantor function
Cantor function

In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous function, but not absolutely continuous....
.

This distribution has neither a probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 nor a probability mass function
Probability mass function

In probability theory, a probability mass function is a function that gives the probability that a discrete random variable random variable is exactly equal to some value....
, as it is not absolutely continuous
Absolute continuity

In mathematics, absolute continuity is a smoothness property which is stricter than continuity and uniform continuity. Both absolute continuity of functions and absolute continuity of measures are defined....
 with respect to Lebesgue measure
Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
, nor has it any point-masses. It is thus neither a discrete nor a continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution
Singular distribution

In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero set where the probability of each point in that set is zero....
.

Its cumulative distribution function is sometimes referred to as the Devil's staircase
Devil's staircase

In mathematics, a singular function is any function ƒ defined on the interval [a, b] that has the following properties:*ƒ is continuous function on [a, b]....
, although that term has a more general meaning.

Characterization


The support of the Cantor distribution is the Cantor set
Cantor set

In mathematics, the Cantor set, introduced by Germany mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties....
, itself the (countably infinite) intersection of the sets

The Cantor distribution is the unique probability distribution for which for any Ct (t ? ), the probability of a particular interval in Ct containing the Cantor-distributed random variable is identically 2-t on each one of the 2t intervals.

Moments

It is easy to see by symmetry that for a random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
 X having this distribution, its expected value
Expected value

In probability theory and statistics, the expected value of a random variable is the Lebesgue integral of the random variable with respect to its probability measure....
 E(X) = 1/2, and that all odd central moments of X are 0.

The law of total variance
Law of total variance

In probability theory, the law of total variance or variance decomposition formula states that if X and Y are random variables on the same probability space, and the variance of X is finite, then...
 can be used to find the variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
 var(X), as follows. For the above set C1, let Y = 0 if X ? [0,1/3], and 1 if X ? [2/3,1]. Then:



From this we get:

A closed form expression for any even central moment
Central moment

In probability theory and statistics, the kth moment about the mean of a real-valued random variable X is the quantity μk := E[k], where E is the expected value....
 can be found by first obtaining the even cumulants

where B2n is the 2nth Bernoulli number
Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
, and then expressing the moments as functions of the cumulants
Cumulant

In probability theory and statistics, if a random variable X admits an expected value ? = E and a variance s2 = E, then these are the first two cumulants: ? = ?1 and s2 = ?2....
.

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