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Encyclopedia > Cantor distribution

The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution is not absolutely continuous with respect to Lebesgue measure, so it has no probability density function; neither is it discrete, since it has no point-masses; nor is it even a mixture of a discrete probability distribution with one that has a density function. It may be characterized by an infinite sequence of coin-tosses in the following way:

Let the random variable X be in the interval [0, 1/3] if "heads" eventuates on the first coin-toss and in the interval [2/3, 1] if "tails.

Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".

Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".

Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".

et cetera, ad infinitum! Then the probability distribution of X is the Cantor distribution.

It is easy to see by symmetry that the expected value of X is E(X) = 1/2.

The law of total variance can be used to find the variance var(X), as follows. Let Y = 1 or 0 according as "heads" or "tails" appears on the first coin-toss. Then:

From this we get: Results from FactBites:

 Cantor function - Wikipedia, the free encyclopedia (476 words) It has no derivative at any member of the Cantor set; it is constant on intervals of the form (0. Extended to the left with value 0 and to the right with value 1, it is the cumulative probability distribution function of a random variable that is uniformly distributed on the Cantor set. The Cantor function is the standard example of what is sometimes called a devil's staircase.
 Cantor distribution - definition of Cantor distribution in Encyclopedia (267 words) The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution is not absolutely continuous with respect to Lebesgue measure, so it has no probability density function; neither is it discrete, since it has no point-masses; nor is it even a mixture of a discrete probability distribution with one that has a density function. Then the probability distribution of X is the Cantor distribution.
More results at FactBites »

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