Definition:Degenerate Distribution

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

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


 * $$\Omega_X = \left\{{r}\right\}$$

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