I'm trying to understand random variables a bit better. As I spent time thinking about, it occurred to me that the probability density of a random variable could itself be considered a random variable.
My reasoning for this is as follows:
A random variable is a function mapping each element of the sample space to the reals.
A probability density is a function mapping each element from the sample space of the random variable to the reals.
This implies that the probability density of a random variable is itself a random variable.
Is this correct?
Edit: Ok, I figured out what the thought process behind the question was. Let us that we have a random variable $X$ with an associated PDF $f_X(x)$. Clearly $f_X(x)$ is not random. However, we could define a new random variable $Y = f_X(X)$. Then $Y$ is a random variable, since it is a function of the random variable $X$. Essentially, the situation is analogous to that between the CDF $F_X(x)$ and the probability integral transform $Y = F_X(X)$