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Information about Cantor Distribution

Cantor
Probability mass function
Cumulative distribution function
Enlarge picture
Cumulative distribution function of the Cantor distribution
Parametersnone
SupportCantor set
Probability mass function (pmf)none
Cumulative distribution function (cdf)Cantor function
Mean1/2
Mediananywhere in [1/3, 2/3]
Moden/a
Variance1/8
Skewness0
Excess kurtosis-8/5
Entropy
Moment-generating function (mgf)
Characteristic function


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, as it is not absolutely continuous with respect to Lebesgue measure, 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.

Its cumulative distribution function 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 (countably infinite) intersection of the sets



The Cantor distribution is the unique probability distribution for which for any Ct (t ∈ { 0, 1, 2, 3, ... }), 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 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 can be used to find the variance 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 can be found by first obtaining the even cumulants[1]



where B2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

External links

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In mathematics, a support of a function f  from a set X  to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
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In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883[1], is a set of points lying on a single line segment that has a number of remarkable and deep properties.
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probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function (abbreviated pdf
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,[1] completely describes the probability distribution of a real-valued random variable X.
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In mathematics, the Cantor function, named after Georg Cantor, is a function c : [0,1] → [0,1] defined as follows:
  1. Express x in base 3. If possible, use no 1s. (This makes a difference only if the expansion ends in 022222... = 100000... or 200000...

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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
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In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
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kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
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Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
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In probability theory and statistics, the moment-generating function of a random variable X is



wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:


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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,[1] completely describes the probability distribution of a real-valued random variable X.
..... Click the link for more information.
In mathematics, the Cantor function, named after Georg Cantor, is a function c : [0,1] → [0,1] defined as follows:
  1. Express x in base 3. If possible, use no 1s. (This makes a difference only if the expansion ends in 022222... = 100000... or 200000...

..... Click the link for more information.
In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
..... Click the link for more information.
probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function (abbreviated pdf
..... Click the link for more information.
In mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected.

Absolute continuity of functions

Definition

Let (X, d) be a metric space and let
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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. It is used throughout real analysis, in particular to define Lebesgue integration.
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In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. Such distributions are not absolutely continuous with respect to Lebesgue measure.
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Listed summits of Aonach Eagach
Name Grid ref Height Status
Meall Dearg NN161583 951 m (3,120 ft) Munro

The Aonach Eagach (IPA: [ɯnəx egəx]
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In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883[1], is a set of points lying on a single line segment that has a number of remarkable and deep properties.
..... Click the link for more information.
A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
..... Click the link for more information.
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


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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
..... Click the link for more information.
In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k
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cumulants: μ = κ1 and σ2 = κ2.

The cumulants κn are defined by the cumulant-generating function:


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