What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean? This is a bit of a silly question, but I can't seem to find the answer anywhere.
I feel like $X\sim \mathcal{N}(\mu,\sigma^{2})$ means that $\sim$ is a relation, but if it is a relation, what precisely is this relation?
If this is a relation, could I instead write $X\in \mathcal{N}(\mu,\sigma^{2})/ \sim$?
One definition I think would be reasonable is to say that $X \sim Y$ if $X$ and $Y$ have the same characteristic or moment generating function, but it seems a little heavy handed to be the "right" definition.
Another definition I would guess is to put a topology on something which characterizes the random variables and say $X\sim Y$ if they satisfy a homeomorphism?
From the homeomorphism viewpoint, I would want to conclude that if $X$ is a random variable whose image is always non-negative, then $X^{2}$ could be normally distributed since you can take the positive square root and exhibit a bijection between $X$ and $X^{2}$ without going into $\mathbb{C}$, but I know this to not be true since it is Chi-Squared, so this can't be a reasonable definition.

It means "distributed like". It's just a shorthand way of saying "$X$ assumes a normal distribution with these parameters."
