notation for two random variables with the same distribution

Question

Suppose $X$ and $Y$ have the same distribution, can I write $X \sim Y$ or is there some other notation for this?

Using tilde feels a bit strange since usually you have $X \sim N(0, 1)$.

Answer 1

In general, you can write $$X\stackrel d= Y,$$ as@eloiPrime suggested. If for example $X$ and $Y$ are standard normal random variables, then you could write $$X,Y\sim\mathcal N(0,1).$$ If in addition $X$ and $Y$ are independent, then you could write $$X,Y\stackrel{i.i.d.}\sim\mathcal N(0,1).$$

Answer 2

You should write $$X \stackrel{d}{=} Y.$$ This means that the random variables are equal in distribution. The tilde notation would imply that $X$ is distributed as $Y$, which seems rather awkward.