Let $X$ be a compound binomial random variable (i.e. having a compound binomial distribution):
$$X\sim\text{Binomial}(n,P)\tag1$$ where $n\in\mathbb{N}$ is fixed and the success probability $P$ is some random variable. As I understand, (1) is equivalent to
$$X|(P=p)\sim\text{Binomial}(n,p)\tag2$$
Question: does it make sense to write
$$X|P\sim\text{Binomial}(n,P)\tag3$$
- Given (1) I am hesitant to write (3), but since (2) is equivalent to (1) it feels like (3) is just a shorter notation for (2). Please, let me know if I am wrong here!
- Some people use this "standalone" notation like $X|P$, some only use it under conditional expectation and I feel like they are deliberately trying to avoid introducing it, so I wonder how one should do/write it correctly...
I would actually not write $(1)$, for the reason that capitalization of letters to denote random variables is not a universally followed practice and it is not necessarily a reliable visual indicator that $X$ obeys a hierarchical model. Another reason for not using $(1)$ is that it becomes unclear how you would specify the unconditional or marginal distribution of $X$. For instance, if $P$ obeys a beta distribution, then the marginal distribution is called beta-binomial. Specifically, I would write such a hierarchical model as follows:
$$P \sim \operatorname{Beta}(\alpha,\beta), \\ X \mid P \sim \operatorname{Binomial}(n, P), \\ X \sim \operatorname{BetaBinomial}(n, \alpha, \beta).$$
Note how we have clearly distinguished the conditional distribution of $X$ given $P$, which is binomial, versus the unconditional/marginal distribution of $X$, which is beta-binomial (and no longer depends on $P$).
For these two major reasons, I would say at best $(1)$ is imprecise, and at worst, incorrect.
$(2)$ and $(3)$ are conceptually equivalent, much in the way that we regard something like $$\operatorname{E}[X \mid Y]$$ as a function of the random variable $Y$, and $$\operatorname{E}[X \mid Y = y]$$ as a function of the realization $y$.