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.[1]
Other names
[edit]These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous.[1]
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.[1]
Example
[edit]An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Less curious examples appear in higher dimensions. For example, the upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
[edit]References
[edit]- ^ a b c "Singular distribution - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-08-23.
External links
[edit]
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