Degenerate distribution

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Degenerate
Probability mass function
Plot of the degenerate distribution PMF for k0=0
PMF for k0=0. The horizontal axis is the index k. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Plot of the degenerate distribution CDF for k0=0
CDF for k0=0. The horizontal axis is the index k.
Parameters k_0 \in (-\infty,\infty)\,
Support k=k_0\,
Probability mass function (pmf) \begin{matrix} 1 & \mbox{for }k=k_0 \\0 & \mbox{otherwise } \end{matrix}
Cumulative distribution function (cdf) \begin{matrix} 0 & \mbox{for }k<k_0 \\1 & \mbox{for }k>k_0 \end{matrix}
Mean k_0\,
Median N/A
Mode k_0\,
Variance 0\,
Skewness 0\,
Excess kurtosis 0\,
Entropy 0\,
Moment-generating function (mgf) e^{k_0t}\,
Characteristic function e^{ik_0t}\,

In mathematics, a degenerate distribution is the probability distribution of a discrete random variable that assigns all of the probability, i.e. probability 1, to a single number, a single point, or otherwise to just one outcome of a random experiment. Examples are a two-headed coin, a die that always comes up six. This does not sound very random, but it satisfies the definition of random variable.

The degenerate distribution is localized at a point k0 in the real line. The probability mass function is given by:

f(k;k_0)=\left\{\begin{matrix} 1, & \mbox{if }k=k_0 \\ 0, & \mbox{if }k \ne k_0 \end{matrix}\right.

The cumulative distribution function of the degenerate distribution is then:

F(k;k_0)=\left\{\begin{matrix} 1, & \mbox{if }k\ge k_0 \\ 0, & \mbox{if }k<k_0 \end{matrix}\right.

There can be some ambiguity in the value of the cumulative distribution function at k = k0. In the above case the convention F(k0;k0) = 1 has been chosen.

[edit] Status of its PDF

As a discrete distribution, the degenerate distribution does not have a density.

The degenerate distribution of a continuous variable is described by the Dirac delta function.


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