Usually a probability distribution over discrete variables is described using a probability mass function (PMF):

When working with continuous random variables, we describe probability distributions using a probability density function (PDF) rather than a probability mass function.

-- Deep Learning by Goodfellow, Bengio, and Courville

However, Wolfram Mathworld is using PDF to describe the probability distribution over discrete variables:

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Is this a mistake? or it does not much matter?

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    $\begingroup$ That's sloppy, in my opinion, but not very important. It's even defensible if they approach probability from the standpoint of measure theory, though that seems like a bit much for an introduction to flipping a coin. (Weird enough, they don't appear to have an article on PMFs.) $\endgroup$ – Dave Jul 28 '19 at 23:39
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    $\begingroup$ a pmf is a density against the counting measure $\endgroup$ – Xi'an Jul 29 '19 at 1:34
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    $\begingroup$ When you discuss the probability theory at the level of measure space specified by 3 elements, pdf and pmf have no different, so the pmf is dropped. All distributions can be specified by pdf. wolfram is a math website, so it is not surprise that they use high level math to talk about probability. Here is good free reading. stat.washington.edu/~pdhoff/courses/581/LectureNotes/… $\endgroup$ – user158565 Jul 29 '19 at 2:06

It is not a mistake: In the formal treatment of probability, via measure theory, a probability density function is a derivative of the probability measure of interest, taken with respect to a "dominating measure" (also called a "reference measure"). For discrete distributions over the integers, the probability mass function is a density function with respect to counting measure. Since a probability mass function is a particular type of probability density function, you will sometimes find references like this that refer to it as a density function, and they are not wrong to refer to it this way.

In ordinary discourse on probability and statistics, one often avoids this terminology, and draws a distinction between "mass functions" (for discrete random variables) and "density functions" (for continuous random variables), in order to distinguish discrete and continuous distributions. In other contexts, where one is stating holistic aspects of probability, it is often better to ignore the distinction and refer to both as "density functions".

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  • $\begingroup$ Thanks for your answer. Does treatment "In the formal treatment of probability" mean notation, perspective, convention or something else? $\endgroup$ – czlsws Jul 29 '19 at 7:13
  • $\begingroup$ When I talk here about the "formal treatment" I am referring to the modern basis of probability theory, which is a subset of measure theory. That is the mathematical theory that is accepted as the formal underpinning of probability. $\endgroup$ – Ben Jul 29 '19 at 7:23
  • $\begingroup$ "a probability density function is a derivative of the probability measure of interest" It seems to me that in some sense it's more of an "anti-integral" than a derivative. There are discontinuous PDFs, such as the uniform distribution, and discrete distributions can be treated as being sums of Dirac delta functions. In those cases, one would have to have to generalize the concept of a derivative far beyond the ordinary understanding for it to apply. $\endgroup$ – Acccumulation Jul 29 '19 at 16:37
  • $\begingroup$ @Acccumulation - how is the uniform distribution discontinuous? ... and measure theory is a far more general treatment of integration and differentiation than the ordinary understanding of Calc I and II provides. $\endgroup$ – jbowman Jul 29 '19 at 19:33
  • $\begingroup$ @Accumulation: Yes, that's a fair characterisation, and indeed, that is what is done. Technically the density is a Radon-Nikodym derivative, which is indeed a type of "anti-integral" of the type you describe. $\endgroup$ – Ben Jul 29 '19 at 22:09

In addition to the more theoretical answer in terms of measure theory, it is also convenient to not distinguish between pmfs and pdfs in statistical programming. For example, R has a wealth of built-in distributions. For each distribution, it has 4 functions. For example, for the normal distribution (from the help file):

dnorm gives the density, pnorm gives the distribution function, qnorm gives the quantile function, and rnorm generates random deviates.

R users rapidly become use to the d,p,q,r prefixes. It would be annoying if you had to do something like drop d and use m for e.g. the binomial distribution. Instead, everything is as an R user would expect:

dbinom gives the density, pbinom gives the distribution function, qbinom gives the quantile function and rbinom generates random deviates.
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    $\begingroup$ scipy.stats distinguishes, some objects have a pdf method and others have a pmf method. It really annoys me! $\endgroup$ – Matthew Drury Jul 29 '19 at 16:19

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