web.archive.org

Cantor distribution: Information and Much More from Answers.com

️Thu Jul 23 1998

Cantor

Probability mass function
Cumulative distribution function Cumulative distribution function of the Cantor distribution
Parameters	none
Support	Cantor set
Probability mass function (pmf)	none
Cumulative distribution function (cdf)	Cantor function
Mean	1/2
Median	anywhere in [1/3, 2/3]
Mode	n/a
Variance	1/8
Skewness	0
Excess kurtosis	-8/5
Entropy
Moment-generating function (mgf)	$e^{t/2} \prod_{i=1}^{\infty} \cosh{\left(\frac{t}{3^{i}} \right)}$
Characteristic function	$e^{\mathrm{i}\,t/2} \prod_{i=1}^{\infty} \cos{\left(\frac{t}{3^{i}} \right)}$

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

Failed to parse (unknown function\begin): \begin{align} C_{0} = & [0,1] \\ C_{1} = & [0,1/3]\cup[2/3,1] \\ C_{2} = & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\ C_{3} = & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\ & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\ C_{4} = & \cdots . \end{align}

The Cantor distribution is the unique probability distribution for which for any C_t (t ∈ { 0, 1, 2, 3, ... }), 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 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:

Failed to parse (unknown function\begin): \begin{align} \operatorname{var}(X) & = \operatorname{E}(\operatorname{var}(X\mid Y)) + \operatorname{var}(\operatorname{E}(X\mid Y)) \\ & = \frac{1}{9}\operatorname{var}(X) + \operatorname{var} \left\{ \begin{matrix} 1/6 & \mbox{with probability}\ 1/2 \\ 5/6 & \mbox{with probability}\ 1/2 \end{matrix} \right\} \\ & = \frac{1}{9}\operatorname{var}(X) + \frac{1}{9} \end{align}

From this we get:

$\operatorname{var}(X)=\frac{1}{8}.$

A closed form expression for any even central moment can be found by first obtaining the even cumulants[1]

$\kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}} {n (3^{2n}-1)},$

where B_2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

External links

Morrison, Kent. "Random Walks with Decreasing Steps", Department of Mathematics, California Polytechnic State University, 1998-07-23. Retrieved on 2007-02-16.

	Probability distributions []
Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • Infinite divisibility (probability) • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)