# What is a good name for a density function that does not relate to probability?

There is confusion between normalized functions whose area under the curve is one, i.e., density functions, and probability density functions that are not only density functions but that are measures of probability per unit area. One can have lots of other things per unit area, like concentration. Currently, some authors refer to such density functions as pdf's despite the confusion that this causes. One work around is to just use $$f(t)$$ or of $$f(x)$$ or whatever and say that it is normalized. However, the rules for how to use density functions are so well documented for statistics, and just because we are not using just probability models does not mean that we are not using statistics; it is just that not all statistics are probabilities.

It would not be a good idea to call generalized density functions df, as df is used for degrees of freedom. Any ideas here are welcome, odf ordinary density functions, gdf, general density function, nf, normalized function. Not a clue what should be done, but something is needed because a pdf is often confused with randomness even though as an $$f(x)$$ all it is, is a model, a formula, a shell, that sure can be used as a model for a random variable, but as a model it is not itself random, it's just a function.

• @whuber This is at the root of what is bothering me. – Carl Apr 22 '19 at 23:09
• Any almost-everywhere-nonnegative Borel-measurable function on $\mathbb{R}$ that integrates to 1 is the probability density function of some random variable, so in my opinion there is no harm calling such functions "probability density functions" – Artem Mavrin Apr 22 '19 at 23:43
• One speaks of mass density functions or energy density functions or populatoin density functions, etc. – Michael Hardy Apr 23 '19 at 0:42
• Who exactly does call functions that are not related to probability as "probability density functions"? Could you give any examples? – Tim Apr 27 '19 at 20:35
• @Tim Lots of people do. Cheng H, Gillespie WR, Jusko WJ. Mean residence time concepts for non‐linear pharmacokinetic systems. Biopharm Drug Dispos. 1994;15:627-41. For example, Cheng et al. start by hand waving by saying that molecules are random therefore we should model concentrations with probability functions, which is nonsense. How much baloney do you want to see? – Carl Apr 28 '19 at 11:40

• @Carl : Yes, probabilities are extensive, but probability DENSITIES are intensive. $\qquad$ – Michael Hardy Oct 4 '19 at 20:29
• @Carl : If I wanted to give an example of the ratio of two extensive quantities being intensive, I would mention the one that I suspect most people see in about eight grade: $\text{miles} \div \text{hours} = \text{miles per hour},$ i.e. $\text{distannce}/\text{time} = \text{rate, or speed}. \qquad$ – Michael Hardy Oct 4 '19 at 20:35