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.
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.
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 distributedrandom variables (the convolution of two uniform distributions).
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