Degenerate distribution

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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 <math>k_0</math> in the real line. The probability mass function is given by:

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

The cumulative distribution function of the degenerate distribution is then:

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

There can be some ambiguity in the value of the cumulative distribution function at <math>k=k_0</math>. In the above case the convention <math>F(k_0;k_0)=1</math> has been chosen.

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