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When people are talking about a probability distribution, does this include generalized functions such as the Dirac-delta distribution or not?

Definition of probability distribution:

A probability distribution is a list of all of the possible outcomes of a random variable along with their corresponding probability values.

So a probability distribution is basically a mapping between outcomes and their probability.

My understanding: Special cases of probability distribution include PDF (when the r.v. is continuous), probability mass function (when the r.v. is discrete), and also the Dirac-delta distribution. A "probability distribution" must be a distribution in all cases. This is straightforward. Math praises precise language: by using the same word, "distribution" must be a "distribution", and a "probability distribution" must be a special case of distribution.

Though, I found many internet sources that argue against my point, so please correct me if I am wrong. I also found no formal sources linking "probability distribution" with the "distribution" (a.k.a. generalized function); any help on this will be very helpful!

Note1: some researchers defined a term "generalized probability density function" which does not necessarily related to this question.

Note2: a PDF is a distribution. But some distributions, such as the Dirac-delta distribution cannot be a PDF, see: Can a Dirac delta function be a probability density function of a random variable?.

dodo
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3 Answers3

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The term "probability distribution" is indeed often used in a loose sense in probability theory and the definition you give is somehow limited. This definition corresponds to the law of a random variable but we can speak of a probability distribution without reference to a random variable, just by defining it as a positive measure on the Borel $\sigma$-algebra of the set ${\Bbb R}$.

It is true that all probability measure on ${\Bbb R}$ is a distribution in the sense of Schwartz, and measures are exactly distributions that are positive. So the Dirac measure is both a probability measure and a Schwartz distribution.

LL 3.14
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coudy
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Pedantically speaking, a probability distribution induces a "distribution in the sense of distribution theory". The associated distribution is $f \mapsto \int f d\mu$. But this mapping is always a distribution (even if $\mu$ doesn't have finite expectation).

Ian
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The two terms are not directly related, but I think they come from the same origin, which is the use in physics : one is trying to know how a quantity is distributed in space for example.

The Schwartz distributions are more general. They don'thave to be positive or normalized and include for example the derivative of the Dirac delta. Probability distributions on the other hand are positive, and so by a famous theorem, are measures, or distributions of order $0$. It includes the Dirac delta but not its derivative.

LL 3.14
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