> 1) Abandon tradition and write P(X). Indeed this random variable usage may
> not be all that traditional anyway. It seems it may be something started by
> us fairly recently. Looking in other (I admit only the few I own) texts, it
> is used only for real-valued functions in Fisherian texts.
I suspect this is wrong (depending on what you consider recent).
LJ Savage wrote in 1954 Fnd of Statistics (making no distinction
between real & discrete):
"Experts are fairly well agreed on the following definition. A random
variable is a function x attaching a value x(s) in some set X to
every s in a set S on which a probability measure P is defined."
No doubt, he was unduly optimistic.
> 2) Stick with formally defining X as a random variable, but then note that
> in Bayesian applications that this is only true implicitly, that actually
> we indentify X and its values directly. So in practice X becomes more of an
> actual variable, and we say things like `manipulating X'.
It seems to me unobjectionable to talk about either manipulating variables or
manipulating functions. If my dog's barking behavior
(f=doggie-barking-behavior-generator) is a function of its input, I can
manipulate it by presenting it with the right inputs. And, indeed,
I do.
Regards, Kevin