What's the difference between $y(x_1 \vert x_2)$ and $y(x_1 ; x_2)$?

Question

In mathematical notation, what's the difference between $y(x_1 \vert x_2)$ and $y(x_1 ; x_2)$? What's the correct way to read them? Please, notice that the only characters that differ are "|" and ";".

Answer 1

I've seen the first one used in conditional PDFs.

https://en.wikipedia.org/wiki/Conditional_probability_distribution

The first notation $y(x_1 | x_2)$ ... it's just a way to add some extra meaning to the $y(x_1, x_2)$ notation. It just tries to make it easier for the reader to link the meaning of $y$ with the notation $y(x_1|x_2)$ itself. In the case of conditional PDFs this $|$ symbol aims to remind the reader that the word is not just about any PDF $y(x_1, x_2)$ but about a conditional PDF $y(x_1 | x_2)$.

That's all, just a notation thing.

In other contexts, the $|$ symbol may mean something completely different, I guess.

Again, if we specially talk about probabilities and statistics... then $y(x_1, x_2)$ usually means joint PDF while $y(x_1 | x_2)$ means conditional PDF.

https://en.wikipedia.org/wiki/Joint_probability_distribution