A singular distribution or singular continuous distribution is a
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
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
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
.
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. 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 In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero o ...
*
Lebesgue's decomposition theorem
References
External links
Singular distributionin the ''
Encyclopedia of Mathematics''
Types of probability distributions
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