What are meaning of these (P(x;y), P(x;y,z),P(x,y;z))?

Question

I was reading a machine learning book that it uses probabilities like these:

P(x;y), P(x;y,z),P(x,y;z)

I couldn't find what they are and how can I read and understand them?

A part of the context I saw one of these probabilities on it:

enter image description here

Answer 1

Note that different distributions have different parameters. Say you have $X \sim \mathcal{N}(\mu,\sigma^2)$. Instead of denoting the distribution as $f_X(x)$ (or $p_X(x)$ for discrete random variables), the book uses $f_X(x;\mu,\sigma)$ (or $f(x;\mu,\sigma)$; simply omitting the subscript). The variables before the semicolon denote the realization of the random variable(s) and ones right after denote the parameters that define such distribution.

EDIT

Regrading your inquiry about the parameters, again let us look at the Gaussian distribution. It is characterized by two parameters, namely: mean (denoted $\mu$) and variance (denoted $\sigma^2$). These parameters alter the shape of the distribution. In particular, the mean defines where the curve is centered at while the variance define the spread or how much the realizations vary around the mean. I got the figure below from here to help you better understand what is it that I'm describing. I hope this clears out your confusion.