Loading AI tools
From Wikipedia, the free encyclopedia
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].
This article relies largely or entirely on a single source. (March 2024) |
These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous[1].
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].
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.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.