Cantor distribution

Cantor

Probability mass function
Cumulative distribution function File:CantorFunction.png 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)	<math>e^{t/2} \prod_{i=1}^{\infty} \cosh{\left(\frac{t}{3^{i
Characteristic function	{{{char}}}

                                               \right)}</math>|
  char       =<math>e^{\mathrm{i}\,t/2} 
                    \prod_{i=1}^{\infty} \cos{\left(\frac{t}{3^{i}}
                                               \right)}</math>|
}}

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

<math>

\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} </math>

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:

<math>

\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} </math>

From this we get:

<math>\operatorname{var}(X)=\frac{1}{8}.</math>

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

<math>

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

</math>

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. 

it:Variabile casuale di Cantor