A degenerate distribution is the probability distribution of a random variable which always has the same value. Examples are a two-headed coin, a die that always comes up six. This doesn't sound very random, but it satisfies the definition of random variable.
The degenerate distribution is localized at a point x in the real line. On this page it is enough to think about the example localized at 0: that is, the unit measure located at 0.
The cumulative distribution function of the degenerate distribution is then the Heaviside step function:
As a discrete distribution, the degenerate distribution does not have a density.
P.A.M. Dirac's delta function can serve this purpose. But a serious theory awaited the invention of distributions by Laurent Schwartz.
NB: There is an unfortunate ambiguity in the meaning of the word distribution. The meaning given to it by Schwartz is not the meaning of the word distribution in probability theory.Status of its PDF
