degenerate distribution

In mathematics, 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:

\theta_0(x)=\left\{\begin{matrix} 1, & \mbox{if }x\ge 0 \\ 0, & \mbox{if }x<0 \end{matrix}\right.

Status of its PDF

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.

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