查看“︁康托尔分布”︁的源代码

{{Probability distribution
| name       = 康托爾
| type       = 質量<!-- not technically correct; added only so template pmf entry would parse correctly; used "mass" rather than "density" since cdf plot suggests it *might* have a pmf -->
| cdf_image  =[[File:CantorEscalier-2.svg|325px|Cumulative distribution function for the Cantor distribution]]|
| parameters = 無
| support    = [[康托爾集]]
| pdf        = 無
| cdf        = [[康托爾函數]]
| mean       = 1/2
| median     = 在 [1/3,&nbsp;2/3] 間的任何數
| mode       = n/a
| variance   = 1/8
| skewness   = 0
| kurtosis   = &minus;8/5
| entropy    = 
| mgf        = <math>e^{t/2} \prod_{k=1}^\infty \cosh\left(\frac{t}{3^k}\right)</math>
| char       = <math>e^{it/2} \prod_{k=1}^\infty \cos\left(\frac{t}{3^k}\right)</math>
}}
'''康托尔分布'''是一种[[累积分布函数]]是[[康托尔函数]]的[[概率分布]]。

该分布即没有[[概率密度函数]]，也没有[[概率质量函数]]，因为虽然其累积分布函数是一个[[连续函数]]，但其分布在[[勒贝格测度]]意义下既不是[[绝对连续]]的，也没有任何点质量。 因此它既不离散的概率分布，也不是一个绝对连续的概率分布，同时不是这两个混合的概率分布。相反，它是一个[[奇异分布]]的例子。

其累积分布函数是处处连续的，但也几乎处处水平，所以有时被称为魔鬼的楼梯，虽然这个用语有更广泛的意义。

== 特征 ==
康托尔分布的基础是[[康托尔集|康托集]]，本身是多个可数无限集的交:

: <math>
\begin{align}
 C_0 = {} & [0,1] \\[8pt]
 C_1 = {} & [0,1/3]\cup[2/3,1] \\[8pt]
 C_2 = {} & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\[8pt]
 C_3 = {} & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\[4pt]
         {} & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\[8pt]
 C_4 = {} & [0,1/81]\cup[2/81,1/27]\cup[2/27,7/81]\cup[8/81,1/9]\cup[2/9,19/81]\cup[20/81,7/27]\cup \\[4pt]
         & [8/27,25/81]\cup[26/81,1/3]\cup[2/3,55/81]\cup[56/81,19/27]\cup[20/27,61/81]\cup \\[4pt]
         & [62/81,21/27]\cup[8/9,73/81]\cup[74/81,25/27]\cup[26/27,79/81]\cup[80/81,1] \\[8pt]
 C_5 = {} & \cdots
\end{align}
</math>

康托尔分布对任何 ''C''<sub>''t''</sub> (''t''&#x20;∈&#x20;{&#x20;0,&#x20;1,&#x20;2,&#x20;3,&#x20;...&#x20;}) 中 2<sup>''t''</sup> 个包含康托尔分布随机变量的特定区间，都有独特的概率 2<sup>-''t''</sup>.

== 矩 ==
通过对称性很容易看出，具有这样分布的一个[[随机变量]] X，其[[期望值]] E(''X'') = 1/2，且所有 X 的奇数阶中心矩都是 0。

[[方差]] var(''X'') 可由{{link-en|总方差定律|Law of total variance}}求得。具体操作如下：对上述集合 ''C''<sub>1</sub>，如果 ''X''&#x20;∈&#x20;[0,1/3] 则令 Y = 0，如果 ''X''&#x20;∈&#x20;[的2/3，1]，令 Y = 1。然后有

: <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>

从而我们得到：

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

任意偶数阶中心矩的封闭表达式可由：先获得偶数项[[累积量]][http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf] {{Wayback|url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |date=20151202055102 }}

:<math>
 \kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}}
                    {n\, (3^{2n}-1)}, \,\!
</math>
其中 ''B''<sub>2''n''</sub> 是 第2''n ''个 [[伯努利数]]，然后用该累积量的方程作为矩的表达。

== 参考文献 ==

* {{Cite book|title=Geometry of Fractal Sets|last=Falconer|first=K. J.|publisher=Cambridge Univ Press|year=1985|location=Cambridge & New York}}
* {{Cite book|title=Real and Abstract Analysis|url=https://archive.org/details/realabstractanal00hewi_0|last=Hewitt|first=E.|last2=Stromberg|first2=K.|publisher=Springer-Verlag|year=1965|location=Berlin-Heidelberg-New York}}
* {{Cite news|title=Fourier Asymptotics of Cantor Type Measures at Infinity|last=Hu|first=Tian-You|work=Proc. A.M.S.|last2=Lau|first2=Ka Sing|year=2002|issue=9|volume=130|pages=2711–2717}}
* {{Cite book|title=Probability Theory & Stochastic Processes|last=Knill|first=O.|publisher=Overseas Press|year=2006|location=India}}

* {{Cite book|title=The Fractal Geometry of Nature|url=https://archive.org/details/fractalgeometryo00beno|last=Mandelbrot|first=B.|publisher=WH Freeman & Co.|year=1982|location=San Francisco, CA}}
* {{cite book|title=Geometry of Sets in Euclidean Spaces|last=Mattilla|first=P.|publisher=Cambridge University Press|year=1995|location=San Francisco}}

* {{cite book|title=Theory of the Integral|last=Saks|first=Stanislaw|publisher=PAN|year=1933|location=Warsaw}} (Reprinted by Dover Publications, Mineola, NY.

== 外部链接 ==

* {{Cite web|url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf|title=Random Walks with Decreasing Steps|accessdate=2007-02-16|date=1998-07-23|last=Morrison|first=Kent|publisher=Department of Mathematics, California Polytechnic State University|archive-date=2015-12-02|archive-url=https://web.archive.org/web/20151202055102/http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf|dead-url=yes}}
[[Category:连续分布]]