Clarifying Probability Mass Function (PMF)

Question

I am currently reading Deep Learning book, and I want to get better understanding of probability theory. In chapter 3.3.1 of Deep Learning book it states that:

Often we associate each random variable with a different probability mass function and the reader must infer which PMF to use based on the identity the random variable, rather than on the name of the function;P(x) is usually not the same as P (y).

And not many paragraphs latter, it saids the following:

Probability mass functions can act on many variables at the same time. Such a probability distribution over many variables is known as a joint probability distribution.P(x=x, y=y) denotes the probability that x=x and y=y simultaneously. We may also write P (x, y) for brevity

I am having hard time grasping these two paragraphs. What do they mean when they say that P(x) is USUALLY not the same as P(y). As I understood, random variable is basically a random phenomenon from a real world that we wish to model. And each random phenomenon has its own Probability Mass function. Does this mean that the first paragraph indicates that random variable y represents a different phenomenon, and in the second paragraph random variable y represents the same type of phenomenon as x and that is why we use the same probability mass function?

Thanks in advance!

Answer 1

I believe they mean the following: "We will use $P$ to denote the concept of a probability mass function (pmf). $P(x)$ is the pmf of random variable $x$, while $P(y)$ is the pmf of random variable $y$. The two variables are usually the results of different random experiments, hence we use two separate letters, $x$ and $y$. As such, they will probably have different pmfs. But somethimes, it might happen that they have the same."

I think the essential source of confusion is that $P$ is not used to denote any single pmf, but is the name used to refer to a pmf. As a parallel, height_of(Stefan) and height_of(Mark) may be different, or they may be the same. height_of is just the name of the thing we are measuring, not the thing itself.

P(x,y) is thus the pmf of the joint variable (x,y). P((x, y)) would be a more unambiguous notation, but I guess is omitted for brevity. If you need more clarification on joint variables let me know. I hope this helps.