“Cantor distribution”的意思、由来-开放百科全书

词条

Cantor distribution

释义

Characterization
Moments
References
External links

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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 its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have 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:

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:

From this we get:

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

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

References

{{cite book |first=K. J. |last=Falconer |title=Geometry of Fractal Sets |publisher=Cambridge Univ Press |location=Cambridge & New York |year=1985}}
{{cite book |first1=E. |last1=Hewitt |first2=K. |last2=Stromberg |title=Real and Abstract Analysis |publisher=Springer-Verlag |location=Berlin-Heidelberg-New York |year=1965}}
{{cite news |first1=Tian-You |last1=Hu |first2=Ka Sing |last2=Lau |title=Fourier Asymptotics of Cantor Type Measures at Infinity |journal=Proc. A.M.S. |volume=130 |number=9 |year=2002 |pages=2711–2717}}
{{cite book |first=O. |last=Knill |title=Probability Theory & Stochastic Processes |publisher=Overseas Press |location=India |year=2006}}
{{cite book |first=B. |last=Mandelbrot |title=The Fractal Geometry of Nature |publisher=WH Freeman & Co. |location=San Francisco, CA |year=1982}}
{{cite book |first=P. |last=Mattilla |title=Geometry of Sets in Euclidean Spaces |publisher=Cambridge University Press |location=San Francisco |year=1995}}
{{cite book |first=Stanislaw |last=Saks |title=Theory of the Integral |publisher=PAN |location=Warsaw |year=1933}} (Reprinted by Dover Publications, Mineola, NY.

External links

{{cite web |last=Morrison |first=Kent |url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |title=Random Walks with Decreasing Steps |publisher=Department of Mathematics, California Polytechnic State University |date=1998-07-23 |accessdate=2007-02-16}}

2 : Continuous distributions|Georg Cantor

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