What does $x \sim X$ mean in probability

Question

I just want to confirm what $x \sim X$ mean in probability. What does the small $x$ and big $X$ represent? And can we replace $\sim$ with $=$.

Disclosure: I had read many probability textbooks from front to back such as Leon-Garcia and Papoulis, but never encountered this notation until I looked at some notes online. I think this is a notation used in stastistics, because $\sim$ is not very informative in my view.

Answer 1

Typically you would say something like the following

$$ X \sim \textrm{Uniform}(-1,1) \tag{1} $$

then $X$ is a random variable and it follows the uniform distribution with parameters $-1$ and $1$. Typically a uniform distribution is like this.

$$ f_{X}(x) =\begin{align}\begin{cases} \frac{1}{b-a} & \textrm{ for } a \leq x \leq b \\ 0 & \textrm{ everywhere else } \end{cases} \end{align} \tag{2}$$

when with parameters $-1$ and $1$ we have

$$ f_{X}(x) =\begin{align}\begin{cases} \frac{1}{2} & \textrm{ for } -1 \leq x \leq 1 \\ 0 & \textrm{ everywhere else } \end{cases} \end{align} \tag{3}$$

In other words, it is saying that $x$ is a random variable and following a distribution given by $X$ in your case, with a distribution function given by it. In other words, it is like the $=$ sign but typically it can refer to a family of functions.

Edit:

If you read the paper in section $3$ on page $2$ it literally defines things.

To learn the generator’s distribution $p_{g}$ over data x, we define a prior on input noise variables $p_{z}(z)$,

It uses notation like this in $4$

The generator G implicitly defines a probability distribution $p_{g}$ as the distribution of the samples G(z) obtained when $z ∼ p_{z}$

...did you read the paper?

Edits:

For the second part if you read the paper, it is defined on page $2$.

where $p$ is the distribution to learn and $q_{\theta}$ is the distribution defined by the implicit generator. The expectation is minimized over a parametrized class of functions

and you note that $\otimes$ is the tensor product.