The question of best practices for probability notation is a common one on this forum, e.g.here, here, here, and elsewhere, e.g. here, here, here. Despite this, I still have unresolved questions.
As an example, if I have a coin whose probability of coming up Heads is $\theta$, then I would normally say that the outcome of the coin toss is modelled as a binary random variable $X$ whose probability distribution is a Bernoulli distribution with parameter $\theta$, or $$ X \sim \text{Bernoulli}(\theta). $$
My first question is whether or not, even if everyone knows what I mean, this is sloppy notation and I can and should be more precise in either or both my notation or terminology?
A related question is, if I observed $n$ repetitions of coin toss above, I would normally write $$ X_i \sim \text{Bernoulli}(\theta), \quad{\text{for $i \in 1...n$}}. $$ By this, I take it that I am saying that I have $n$ independent random variables, each with one observation. Should I instead be saying that there is actually just one random variable, i.e. $X$, but $n$ independent observations of it? In other words, in a situation like this, is it better to say that we have 1 random variable, or $n$ iid random variables?
