TITLE: "X is distributed as ..." notation
QUESTION [5 upvotes]: A question about notation:
We sometimes use $\sim$ to denote "distributed as" e.g. if $X$ is Gaussian we write $X \sim N(\mu, \sigma^2)$. 
Is it acceptable to use the "~" notation for an arbitrary distribution? e.g. can we write 
$$X \sim \begin{cases} \frac{3}{2}x^2, & x \in [-1,1] \\0 & \text{otherwise} \end{cases}$$
If not, is there another way to write "$X$ is distributed as ..."? i.e. without the more verbose "Let $p_X(x)$ be the pdf of $X$. Then $p_X(x)=$ ...

REPLY [4 votes]: I see no huge problem with the notation though you don't see it a lot. It's reasonably clear what you mean, and that's pretty much the goal with notation.
However I don't think "Let $X$ be a random variable with PDF $p_X(x) = \ldots$" is overly verbose. You could even say "Let $p_X(x) = \ldots$" since $p_X$ and $f_X$ are pretty standard notations for the PDF, but I think the first option has the right balance of clarity and efficiency for my taste.

REPLY [2 votes]: To the best of my knowledge, it is not that rare to see $X\sim f_X(x)$ as a shorthand for $X$ is distributed according to $f_X(x)$. However, I've never encountered in any formal settings your version ($\sim$ followed by brackets etc.).