Why is a pmf called a probability mass function and why is a pdf called a probability density function?

Question

Why is a pmf called a probability mass function and why is a pdf called a probability density function?

What do "mass" and "density" represent for a pmf and pdf? Basically my question is, why use the words mass and density? I'm not getting it. Is it simply just part of the name? I know for a pdf you measure the area under the density function to find probability. But why is it called a "density" function. And what does "mass" even mean in a pmf? Please help.

I'm not sure if there is an reason "mass" was used, but it's just the probability of the point being selected. As for "density", it's analogous with mass density from physics, in that you need to integrate the density over a region to get the mass, in this case, the probability "mass". — Joe, Jul 23 '21 at 18:31
I agree with Joe as to density. The discrete case is by analogy with point masses, again from physics. — saulspatz, Jul 23 '21 at 18:34
Google nGrams seems to confirm my guess that probability density came first and was used because it is not a probability but its integral is. Probability mass then came later for discrete distributions as an equivalent to point masses. — Henry, Jul 23 '21 at 18:35
Why is a cat called a 'cat' and sometimes a 'kitty', but a dog called a 'dog'. Why is a PMF sometimes called a PDF. — BruceET, Jul 24 '21 at 16:47
Does this answer your question? Probability density function vs. probability mass function — , Jan 01 '22 at 05:22

score 3 · Answer 1 · answered Jul 24 '21 at 05:08

To get some intuition, we can draw some parallels between physics and probability science!

In physics, mass is the weight over gravity: $$m=\frac{W}{g}$$ In probability science, the probability of a discrete random variable is: $$P(A)=\frac{n(A)}{n(all)}$$ For example, in the experiment of tossing a fair six-sided die, the probability of getting a $5$ is: $$PMF(X=5)=f(X=5)=P(X=5)=\frac{n(5)}{n(all)}=\frac16$$

So, the weight (W) is analogous to the number of ways an event A can occur (n(A)) and the gravity is analogous to the sample space (n(all)).

In physics, density is the mass over volume: $$\rho = \frac{m}{V}$$

In probability science, the probability of a continuous random variable over some interval is: $$P(a\le X\le b)=\int_a^b f_{X}(x)\, dx$$

For example, let the bus waiting time be uniformly distributed: $X \sim [10,30]$. The probability of waiting between $15$ and $20$ minutes is: $$P(X<15)=\int_{15}^{20} \frac{1}{20}\, dx=\frac1{20}\cdot (20-15)=\frac14=0.25.$$

So, the mass is analogous to the interval ($[a,b]=[15,20]$) and the volume is analogous to the entire range ($[c,d]=[10,30]$).

In general, the term "probability distribution function" can be used to imply either probability mass function (for discrete r.v.) or probability density function (for continuous r.v.).

score 2 · Answer 2 · answered Jul 24 '21 at 04:04

We used to refer to the probability of events as a weighted bias. Possibly a reference to a practice of using lead weights to introduce a bias into gambling dice.

So if a cumulative probability measure has something analogous to 'weight' (or rather mass), then its natural to likewise compare its derivative (when appropriate to take it) to a 'density'.

Why is a pmf called a probability mass function and why is a pdf called a probability density function?

2 Answers2