# Definition:Degenerate Distribution

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## Definition

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

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

- $\Omega_X = \set r$

- $\map \Pr {X = k} = \begin{cases} 1 & : k = r \\ 0 & : k \ne r \end{cases}$

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

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 $0$.

## Also see

- Results about
**the degenerate distribution**can be found here.