< List of probability distributions < Singular distribution

## Singular distribution definition

A **singular distribution **is concentrated on a set of Lebesgue measure zero. This means that the probability of any point in the set is zero. A singularity is a point where a function or a measure is undefined; in the context of probability, a singularity can be thought of as a point where the distribution is undefined. This can happen for a variety of reasons, such as when the distribution is concentrated on a set of points that have zero measure.

An alternate definition of a singular distribution is one that is singular with respect to the Lebesgue measure *and *has no **atoms** [1]. An atom is a set of points that cannot be broken down into smaller sets with probability greater than zero.

A singular distribution can also be described as one that has no absolutely continuous or discrete part [2]. It doesn’t have a probability density function, since the Lebesgue integral of any such function would be zero; it is not a discrete probability distribution (pdf) because each discrete point has a zero probability. Thus, a singular distribution can’t be represented as a sum of a continuous pdf and a discrete probability mass function (PMF). On the other hand, a *regular distribution can *be represented as a sum of a continuous PDF and a discrete PMF. Regular distributions comprise most of the probability distributions you’ll come across in statistics.

Singular distributions are sometimes called *singular continuous distributions*, because their cumulative distribution functions (CDFs) are singular and continuous.

## Singular distribution examples

Singular distributions aren’t that common in probability, but there are a few notable ones:

- The
**Cantor Distribution**is a probability distribution whose CDF is the Cantor Function. The Cantor function is an example of a pathological function that is horizontal almost everywhere yet always climbs upwards. This means that the probability of any point in the Cantor set is zero. - A continuous
**degenerate distribution**is a special case of a singular distribution [4]. It is a singular distribution that is also continuous. **Dirac delta distribution:**The Dirac delta is an element of a set of mathematical objects called*distributions*– so the function is more aptly named a “delta distribution.” It is a singular distribution because its pdf is not integrable [5].

The pdf of a singular distribution can be undefined at some points, but that doesn’t mean a singular distribution has an undefined pdf. For example, the Cauchy distribution is a singular distribution, but its pdf is defined at all points.

## References:

[1] Žitkovic, G. Theory of Probability: Measure theory, classical probability and stochastic analysis. Lecture Notes

[2] Wolpert, R. STA 711: Probability & Measure Theory

[3] Image: CantorEscalier.svg: Theonderivative work: Amirki, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[4] Gentle, J. (2003). Theory of Statistics.

[5] Howard, R. (2012). Dirac Delta and Singular Distributions: The General Non-good Function Case. Conference: International Conference on Engineering and Applied Science At: Beijing.