Is probability distribution always a special case of a distribution?

Question

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!

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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?.

Answer 1

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.

Answer 2

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).