\begin{definition}[Definition:Degenerate Distribution]
Let $X$ be a discrete random variable on a probability space.
Then $X$ has a '''degenerate distribution with parameter $r$''' {{iff}}:
:$\Omega_X = \set r$
:$\map \Pr {X = k} = \begin {cases}
1 & : k = r \\
0 & : k \ne r
\end {cases}$
That is, there is only value that $X$ can take, namely $r$, which it takes with certainty.
{{ExtractTheorem}}
It trivially gives rise to a probability mass function satisfying $\map \Pr \Omega = 1$.
Equally trivially, it has an expectation of $r$ and a variance of $0$.
\end{definition}