Notation for random number drawn from a certain probability distribution

Question

Is there a notation for random numbers that are drawn or belong to a specific probability distribution? For example. I have a some function which is dependant on $x$ and some random number which is drawn from a Normal distribution.

to write it as

$$f(x) = x^{2} + G(\sigma, \mu)$$

seems wrong as this implies the PDF itself is added to $x^{2}$. Can anyone suggest a notation showing I am adding a single (or list of numbers) which belong to a defined PDF.

Answer 1

Formally, you need to write two lines: $$f(x)=x^2+N$$ $$N\sim G(\sigma,\mu)$$ The second line says that $N$ is a sample from the $G(\sigma,\mu)$ distribution.

Answer 2

I'd suggest a different notation than Parcly, such as $Z_x = x^2 + N,$ where I write $Z_x$ rather than $f(x)$ to emphasise that it is a random variable that depends on the constant $x$, rather than a (deterministic) function of $x$.

Edit: you should indeed also write that $N\sim G(\sigma, \mu).$