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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.

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    $\begingroup$ 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" $\endgroup$
    – Artem Mavrin
    Commented Apr 22, 2019 at 23:43
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    $\begingroup$ One speaks of mass density functions or energy density functions or populatoin density functions, etc. $\endgroup$
    – Michael Hardy
    Commented Apr 23, 2019 at 0:42
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    $\begingroup$ @ArtemMavrin - and that is a philosophical issue: a Bayesian prior or posterior density might be said not to represent a probability function but instead a distribution of degree of belief, even if they satisfy Kolmogorov's probability axioms. $\endgroup$
    – Henry
    Commented Apr 23, 2019 at 0:42
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    $\begingroup$ Who exactly does call functions that are not related to probability as "probability density functions"? Could you give any examples? $\endgroup$
    – Tim
    Commented Apr 27, 2019 at 20:35
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    $\begingroup$ @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? $\endgroup$
    – Carl
    Commented Apr 28, 2019 at 11:40

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One speaks of "energy density", "mass density", "population density". Energy, mass, and population are, in the language of physicists, extensive quantities, as is probability: they add up. Their densities, on the other hand are intensive rather than extensive: one evaluates the density at a point and integrates it over sets of points.

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  • $\begingroup$ One of the problems here is that molecules or probabilities are extensive and concentration is intensive, i.e., densities are intensive. The ratio of two extensive variables is an intensive variable. For example, dividing the number of particles by the volume produces the particle density: ρ=N/V; an intensive quantity. $\endgroup$
    – Carl
    Commented Oct 4, 2019 at 20:19
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    $\begingroup$ @Carl : Yes, probabilities are extensive, but probability DENSITIES are intensive. $\qquad$ $\endgroup$
    – Michael Hardy
    Commented Oct 4, 2019 at 20:29
  • $\begingroup$ @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. $\text{distannce}/\text{time} = \text{rate, or speed}. \qquad$ $\endgroup$
    – Michael Hardy
    Commented Oct 4, 2019 at 20:35
  • $\begingroup$ Well, the problem I have is that someone is telling me that individual molecules follow a probability density function when rather the data is concentration data in time, and I am trying to explain that concentration, although intensive, has nothing to do with probability. $\endgroup$
    – Carl
    Commented Oct 4, 2019 at 22:05

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