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.

Other names[edit]

These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous.

Properties[edit]

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.

In general, distributions can be described as a discrete distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or can be decomposed into a mixture of these.

Example[edit]

An example is the Cantor distribution; its cumulative distribution function is a devil's staircase.

See also[edit]

External links[edit]