What do we exactly mean by "density" in Probability Density function (PDF)? [duplicate]

Question

In general density is mass/volume. Also it's used for something like population density, which is population/unit area.

What is significance of word density in PDF?

Answer 1

Short answer: Like in physical density, the probability density is probability/volume.

Long answer: For homogeneous objects, density can be defined as you said, $m/V$, with $m$ denoting mass and $V$ its volume. However, if your object is not homogeneous, the density is a function of the space coordinates within the object: $$ \rho(x, y, z) = \lim_{\Delta V \rightarrow 0} \frac{\Delta m(x, y, z)}{\Delta V} $$ i.e. the mass inside an infinitesimal volume around the given coordinates, divided by that infinitesimal volume. Think of a plum pudding: The density at the raisins is different from the density at dough.

For probability, it is basically the same: $$ f(x, y, z) = \lim_{\Delta V \rightarrow 0} \frac{\Delta F(x, y, z)}{\Delta V} $$ where $f$ is the probability density function (PDF) and $F$ the cumulative density function (CDF), so that $\Delta F$ is the infinitesimal probability in the infinitesimal volume $\Delta V$ in the vicinity of coordinates $(x, y, z)$ in the space over which $F$ is defined.

Now, we happen to live in a physical world with three space dimensions, but we are not limited to defining probabilities just over space. In practice, it is much more common to work with probabilities defined over a single dimension, say, $x$. Then the above simplifies to $$ f(x) = \lim_{\Delta x \rightarrow 0} \frac{\Delta F(x)}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{F(x+\Delta x) - F(x)}{\Delta x} $$ But, of course, depending on your probability model, $F$ and $f$ can be defined over any number of dimensions.

Answer 2

You could see the Radon-Nikodym derivative as a formal definition of a more general notion of density.

It is the ratio of two measures (which have the extensive property, they are additive) defined on the same space.

$$\rho = \frac{d \nu}{d \mu}$$

This ratio makes the one quantity measure $\nu$ of a set $S$ expressible by an integral over the other measure $\mu$ $$\nu(S) = \int_S \rho d \mu$$

Typically the denominator $\mu$ is a measure based on a metric measure like distance, area or volume. This is common for densities in physics like mass density, energy density, charge density, particle density.

With the density of probability the denominator can be more generally another type of variable that does not relate to physical space. Yet, often it is similar in the use of the Euclidean measure or Lebesgue measure. It is just that the variable does not need to be a coordinate in physical space.

Answer 3

For a single continuous random variable, the value of the pdf at the point $t$ tells you the density of the probability mass, measured in units of probability mass per unit length, at the point $t$ on the real line. The density of the probability mass can be different at different points on the real line; it is not quite as facile as the mass/volume prescription of high-school physics.