singular distribution

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.

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.

• Singular measure
• Lebesgue's decomposition theorem

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• [https://encyclopediaofmath.org/wiki/Singular_distribution Singular distribution] in the Encyclopedia of Mathematics

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Category:Types of probability distributions

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