Cantor distribution - Alchetron, The Free Social Encyclopedia

Parameters none pmf none Mean 1/2		Support Cantor set CDF Cantor function Median anywhere in [1/3, 2/3]

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, since although it is a continuous function it is not absolutely continuous with respect to Lebesgue measure, nor has it 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 0 = [ 0 , 1 ] C 1 = [ 0 , 1 / 3 ] ∪ [ 2 / 3 , 1 ] C 2 = [ 0 , 1 / 9 ] ∪ [ 2 / 9 , 1 / 3 ] ∪ [ 2 / 3 , 7 / 9 ] ∪ [ 8 / 9 , 1 ] C 3 = [ 0 , 1 / 27 ] ∪ [ 2 / 27 , 1 / 9 ] ∪ [ 2 / 9 , 7 / 27 ] ∪ [ 8 / 27 , 1 / 3 ] ∪ [ 2 / 3 , 19 / 27 ] ∪ [ 20 / 27 , 7 / 9 ] ∪ [ 8 / 9 , 25 / 27 ] ∪ [ 26 / 27 , 1 ] C 4 = [ 0 , 1 / 81 ] ∪ [ 2 / 81 , 1 / 27 ] ∪ [ 2 / 27 , 7 / 81 ] ∪ [ 8 / 81 , 1 / 9 ] ∪ [ 2 / 9 , 19 / 81 ] ∪ [ 20 / 81 , 7 / 27 ] ∪ [ 8 / 27 , 25 / 81 ] ∪ [ 26 / 81 , 1 / 3 ] ∪ [ 2 / 3 , 55 / 81 ] ∪ [ 56 / 81 , 19 / 27 ] ∪ [ 20 / 27 , 61 / 81 ] ∪ [ 62 / 81 , 21 / 27 ] ∪ [ 8 / 9 , 73 / 81 ] ∪ [ 74 / 81 , 25 / 27 ] ∪ [ 26 / 27 , 79 / 81 ] ∪ [ 80 / 81 , 1 ] C 5 = . . .

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.

Recently discovered, the Geometric Mean of all reals in the Cantor Set between (0,1] is approximately 0.274974, which is ≈ 75% of the Geometric Mean of all reals in between (0,1].

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 except for the first moment 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:

var ⁡ ( X ) = E ⁡ ( var ⁡ ( X ∣ Y ) ) + var ⁡ ( E ⁡ ( X ∣ Y ) ) = 1 9 var ⁡ ( X ) + var ⁡ { 1 / 6 with probability 1 / 2 5 / 6 with probability 1 / 2 } = 1 9 var ⁡ ( X ) + 1 9

From this we get:

var ⁡ ( X ) = 1 8 .

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

κ 2 n = 2 2 n − 1 ( 2 2 n − 1 ) B 2 n n ( 3 2 n − 1 ) ,

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

References

Cantor distribution Wikipedia

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Contents

Characterization

Moments

References