{{Short description|Distribution concentrated on a set of measure zero}} {{Distinguish|Singular distribution (differential geometry)}} {{onesource|date=March 2024}} A '''singular distribution''' or '''singular continuous distribution''' is a probability distribution concentrated on a set of Lebesgue measure zero, for which the probability of each point in that set is zero.<ref name=":0">{{Cite web |title=Singular distribution - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Singular_distribution |access-date=2024-08-23 |website=encyclopediaofmath.org}}</ref>
==Properties== 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.<ref name=":0" />
==Example== An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Another is the Minkowski's question-mark distribution. Less curious examples appear in higher dimensions. For example, the upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.{{Citation needed|date=June 2025}}
==See also== *Singular measure *Lebesgue's decomposition theorem
==References== {{reflist}}
==External links== *[https://encyclopediaofmath.org/wiki/Singular_distribution Singular distribution] in the ''Encyclopedia of Mathematics''
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{{DEFAULTSORT:Singular Distribution}} Category:Types of probability distributions
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