What is the relationship between probability densities and mass densities?

In statistics we have probability distribution functions which give us likelihood that some random variable ($$X$$) will equal a particular value. Assuming this variable is continuous, the distribution satisfies: $$\int f_{X}(x) \ dx = 1$$

and can be used to obtain the moments of the variable \begin{align} \mu'_i = \int X^i f_x(x) dx \end{align}

In physics we often work with the mass distribution function which describes the distribution of mass. For example the total mass across a system is given by: $$M_{tot} = \int g_M(m) \ dm$$

and like the statistical PDF we can also derive moments of this mass distribution function by integrating across the distribution function: $$\mu'_i = \int m g_M(m) \ dm$$

I am wondering what the connection between these two concepts is and if there is any way to move between them?

1 Answer

As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.

In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).

You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.

Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.