I just want to be sure about this:
If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ actually is and it is just looked at the probability itself) this could rigorously mean, that there is a map $X: \Omega \rightarrow \mathbb{R}$ and that there is a probability space $(\Omega = \mathbb{R}_{>0}, X^{-1}(\mathscr{B}) , P)$ (where $X^{-1}(\mathscr{B})$ is the inverse map of the sigma algebra of the Borel sets) such that $P(\{X^{-1}(E)\}):= \int_\mathbb{R} \chi_E(x) \lambda e^{-\lambda x}dx$?
This is one interpretation that came to my mind by thinking about what it could mean. Could anybody here now explain to me where I am wrong and what it actually means? ( So my goal is to give all the details to the phrase ' a random variable $X$ is exponentially distributed)
Does anybody here have an idea? If there is anything unclear about my question, please let me know.