# Physical meaning of “probability density”

Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. Density for me, is the co-domain of pdfs - a scalar dimension with values from zero to infinity.

For instance, I fit a normal distribution to a probability vector on N values (a probability mass function). This vector contains normalised values of the actual frequency counts in a histogram with N bins. The normalisation is done by dividing each frequency count by the discrete integral "area" of the pmf - so that the pmf values sum to 1. The pmfs now seems to be scaled similarly to the estimated pdfs.

Is probability density a measure of probability? I am sure that the necessary definitions must be hidden somewhere deep down in the guts of measure theory - which is why I have included that as a tag.

-
Just a note. The relevant object is really the measure $p(x)\,dx$ (that you integrate with respect to), not the pdf $p(x)$ itself. For example, under a change of variable $y(x)$, $p(x)$ can be changed almost arbitrarily (multiplied by $1/\frac{dy}{dx}$ which can be almost any function you want), but $p(x)dx=p(y)dy$ stays the same. –  Kirill Jul 2 '13 at 22:32
The units of the codomain of the probability density function must be reciprocal of the units of the domain so that upon integration, we obtain the unitless probability. –  Sammy Black Jul 2 '13 at 22:35

To add to Kirill's comment, you can write $P_0 = P(x_0\leq X \leq x_0+{\rm d}x) = p(x_0)dx$, so the value of the pdf $p(x)$ at $x_0$ is the height of an infinitesimally narrow interval ('slice') of width d$x$ fitting below this function at $x_0$ that will give you a probablity value $P_0$ to find the random variable lying within this interval d$x$. You could also think of it, perhaps more intuitively, in terms of the derivative of the cdf: the pdf at $x_0$ is then the 'speed' or rate, d$P/$d$x$, at which the probability will increase when going from $x_0$ to $x_0+dx$ (for positive d$x$). A "physical" meaning is more difficult, because probability is mathematical... Perhaps if you think of the analogy of "mass density" (~pdf) as a measure of concentration of "mass" (~cdf), that may help?