Singular distribution

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.

Other names

These distributions are sometimes called singular continuous distributions.

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.

Example

An example is the Cantor distribution.

See also

• Singular measure
• Lebesgue's decomposition theorem

External links

• Springer Encyclopaedia of Mathematics