Definition:Degenerate Distribution

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

Then $X$ has a degenerate distribution with parameter $r$ if and only 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.