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!