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Encyclopedia > 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:

## 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.

Results from FactBites:

 Degenerate distribution - definition of Degenerate distribution in Encyclopedia (193 words) In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. As a discrete distribution, the degenerate distribution does not have a density. The meaning given to it by Schwartz is not the meaning of the word distribution in probability theory.
 Probability distribution - Wikipedia, the free encyclopedia (1336 words) A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space. The rectangular distribution is a uniform distribution on [-1/2,1/2]. The triangular distribution on [a, b], a special case of which is the distribution of the sum of two uniformly distributed random variables (the convolution of two uniform distributions).
More results at FactBites »

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