# Let's create [pmf] tag and keep it distinct from [pdf]

I just got clobbered for calling a PMF a PDF. In the tag info it says

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

I have just created the tag. I want this to be more clearly defined at the getgo so I am proposing to keep them separate. The critical difference is that

In case of continuous variables $X$, the PDF $\mathcal{P}_X(x)$ can be integrated over an interval $\mathcal{I}$ (or, more generally, any Borel set) to find the probability that the variable is in that interval:

$$\Pr(X \in \mathcal{I}) = \int_\mathcal{I} \mathcal{P}_X(x) dx.$$

However for the PMF, thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes $x$: $$\sum_{x\in A} f_X(x) = 1$$

I would like to change that currently pmf maps to pdf and these are not actually synonyms.

• +1. I hadn't noticed this use of one tag for two such distinct concepts--even though I was the last to edit it! I guess I just accepted the status quo. It could be a lot of work for someone to go through all the pdf questions (almost 900 of them) to identify which ones really need to be retagged, though!
– whuber Mod
Nov 23, 2017 at 1:05
• @whuber Yup, agreed they are different and I do not like getting clobbered. Moreover, my post defining pmf was deleted, and I cannot tell how this occurred.
– Carl
Nov 23, 2017 at 1:09
• @amoeba What gives with this? Weigh in please.
– Carl
Nov 23, 2017 at 1:10
• Concerning your pmf post, could you provide some details of what you mean by that? Often when stuff disappears it can be recovered, but we need some hints about where to look for it. A few minutes ago I fielded your flag requesting a deletion of one of your answers on the main site, so I did so, but I hope that's not what you're referring to.
– whuber Mod
Nov 23, 2017 at 1:15
• @whuber No, my friend, you did no wrong deletion. I did a pmf tag proposal, but cannot make a wiki tag that is not reviewed as my rep is <5000, was notified that the tag is under review, and it does now not exist. It turns out that gung made the synonym just after I proposed the tag: stats.stackexchange.com/tags/synonyms I would like this discussed, please.
– Carl
Nov 23, 2017 at 3:03
• I've been clobbered here for insisting on not calling a pmf a pdf... Nov 23, 2017 at 3:21
• @Glen_b I would defend your position. A density function is the mass per unit x-axis variable, a mass function has a localized masses at evaluated points on an x-axis like at x=2,3, and are not per unit range.
– Carl
Nov 23, 2017 at 3:30
• I can't see any problem with making [pdf] and [pmf] synonyms. The idea of synonyms is not only about exact synonyms, but about groupping related concepts for easier searching.
– Tim
Nov 23, 2017 at 8:13
• Fashions change, so it's not just one kind of rigour versus its lack. Many modern authors regard density functions as defined widely, it being just a question of what kind of density in what kind of outcome space. So such authors are happy to include functions defined w.r.t. counting measure, that some want to insist are mass functions. See for example Peter Whittle's probability text, various editions and titles from 1970 on. Nov 24, 2017 at 16:19

For the record: our tag description for has always said that one should use it for PMFs too. When you created tag yesterday to replace [pdf] on one of the questions, I rolled your edit back because I thought you are clearly going against our established usage. When you immediately replaced it back, I got a bit annoyed and flagged the post for mod attention suggesting to make pmf the synonym of pdf. @Gung must have seen my flag and decided to act on it.

Now, I agree with Tim that

I can't see any problem with making [pdf] and [pmf] synonyms. The idea of synonyms is not only about exact synonyms, but about groupping related concepts for easier searching.

That said, I think we could keep them separate if we wanted. For this we would need (1) to remove the synonym mapping, and (2) to retag all Qs about PMFs that are tagged [pdf]. It turns out it's not so many: I get only 24 Qs with these two searches: one and two. I expected to find many more, but 24 are easily doable in a couple of days.

I don't have a strong opinion and am fine with it either way.

Personally, I don't see the difference between PDF and PMF particularly important because if one allows delta functions then these two concepts can be combined together. See https://en.wikipedia.org/wiki/Probability_density_function#Link_between_discrete_and_continuous_distributions. As always with delta functions, a little extra care is needed to do it in a mathematically rigorous way (see https://math.stackexchange.com/a/54249), but in practice one does not need to bother because it behaves in an intuitively clear way.

• Comments are not for extended discussion; this conversation has been moved to chat.
– Tim
Nov 28, 2017 at 23:07