Singular distribution

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In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.

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These distributions are sometimes called singular continuous distributions.


Such distributions are not absolutely continuous with respect to Lebesgue measure.

A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero.


An example is the Cantor distribution.

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