Degenerate distribution

Degenerate univariate
Cumulative distribution function CDF for a = 0. The horizontal axis is x.
Parameters	${\displaystyle a\in (-\infty ,\infty )\,}$
Support	${\displaystyle \{a\}}$
PMF	${\displaystyle {\begin{matrix}1&{\mbox{for }}x=a\\0&{\mbox{elsewhere}}\end{matrix}}}$
CDF	${\displaystyle {\begin{matrix}0&{\mbox{for }}x<a\\1&{\mbox{for }}x\geq a\end{matrix}}}$
Mean	${\displaystyle a\,}$
Median	${\displaystyle a\,}$
Mode	${\displaystyle a\,}$
Variance	${\displaystyle 0\,}$
Skewness	undefined
Excess kurtosis	undefined
Entropy	${\displaystyle 0\,}$
MGF	${\displaystyle e^{at}\,}$
CF	${\displaystyle e^{iat}\,}$
PGF	${\displaystyle z^{a}\,}$

In probability theory, a degenerate distribution on a measure space ${\displaystyle (E,{\mathcal {A}},\mu )}$ is a probability distribution whose support is a null set with respect to ${\displaystyle \mu }$. For instance, in the n-dimensional space ℝⁿ endowed with the Lebesgue measure, any distribution concentrated on a d-dimensional subspace with d < n is a degenerate distribution on ℝⁿ. This is essentially the same notion as a singular probability measure, but the term degenerate is typically used when the distribution arises as a limit of (non-degenerate) distributions.

When the support of a degenerate distribution consists of a single point a, this distribution is a Dirac measure in a: it is the distribution of a deterministic random variable equal to a with probability 1. This is a special case of a discrete distribution; its probability mass function equals 1 in a and 0 everywhere else.

In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in a is $${\displaystyle F_{a}(x)=\left\{{\begin{matrix}1,&{\mbox{if }}x\geq a\\0,&{\mbox{if }}x<a\end{matrix}}\right.}$$ Such degenerate distributions often arise as limits of continuous distributions whose variance goes to 0.