Degenerate distribution

In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter definition, it is a deterministic distribution and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of "random variable" even though it does not appear random in the everyday sense of the word; hence it is considered degenerate.

Degenerate univariate

Cumulative distribution function CDF for k0=0. The horizontal axis is x.
Parameters	${\displaystyle k_{0}\in (-\infty ,\infty )\,}$
Support	${\displaystyle x=k_{0}\,}$
PMF	${\displaystyle {\begin{matrix}1&{\mbox{for }}x=k_{0}\\0&{\mbox{elsewhere}}\end{matrix}}}$
CDF	${\displaystyle {\begin{matrix}0&{\mbox{for }}x<k_{0}\\1&{\mbox{for }}x\geq k_{0}\end{matrix}}}$
Mean	${\displaystyle k_{0}\,}$
Median	${\displaystyle k_{0}\,}$
Mode	${\displaystyle k_{0}\,}$
Variance	${\displaystyle 0\,}$
Skewness	undefined
Ex. kurtosis	undefined
Entropy	${\displaystyle 0\,}$
MGF	${\displaystyle e^{k_{0}t}\,}$
CF	${\displaystyle e^{ik_{0}t}\,}$

In the case of a real-valued random variable, the degenerate distribution is a one-point distribution, localized at a point k₀ on the real line. The probability mass function equals 1 at this point and 0 elsewhere.

The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose variance goes to 0 causing the probability density function to be a delta function at k₀, with infinite height there but area equal to 1.

The cumulative distribution function of the univariate degenerate distribution is:

${\displaystyle F_{k_{0}}(x)=\left\{{\begin{matrix}1,&{\mbox{if }}x\geq k_{0}\\0,&{\mbox{if }}x<k_{0}\end{matrix}}\right.}$