Suppose I have a Beta distribution, and want to express in an equation a random sample from this distribution. I could say $x \sim Beta(\alpha, \beta)$. Now, consider the case when $x$ is a constant $k$, plus a random draw from this distribution. How do I write that down? Some ideas I have are:
1) $x = k + \sim Beta(\alpha, \beta)$
2) $x = k + Beta(\alpha, \beta)$
3) $x = k + y$ where $y \sim Beta(\alpha, \beta)$
Which would be the proper way to express this? Or is there another way?
Assuming you like the notation $x \sim Beta(\alpha, \beta)$, then your option (3) is certainly a correct and unambiguous choice. But it is a bit wordy.
I would normally go with option (2) because the reader will always get what it means in context, as long as she realizes that $Beta(\alpha, \beta)$ means a function delivering a random according to the Beta distribution -- and if that is not clear to the reader, then neither will option (3) be clear.