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.
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.
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.
See also Singular measure Lebesgue's decomposition theorem
References
External links Singular distribution in the Encyclopedia of Mathematics