# Definition:Degenerate Distribution

## Definition

Let $X$ be a discrete random variable on a probability space.

Then $X$ has a degenerate distribution with parameter $r$ if:

$\Omega_X = \left\{{r}\right\}$
$\Pr \left({X = k}\right) = \begin{cases} 1 & : k = r \\ 0 & : k \ne r \end{cases}$

That is, there is only value that $X$ can take, namely $r$, which it takes with certainty.

It trivially gives rise to a probability mass function satisfying $\Pr \left({\Omega}\right) = 1$.

Equally trivially, it has an expectation of $r$ and a variance of $0$.

## Also see

• Results about the degenerate distribution can be found here.