After slowly coming to grips with Random Variables and the distributions that they subscribe to (e.g. $X$ ~ $\mathcal{U}(0,1$)) , we introduced the notion of pdf, which I believe I understand in essence, but I am rather confused when, for example, we are told that for the exponential distribution to parameter $\lambda > 0$
the density $f$ can be said to be $f(x):=\lambda e^{-\lambda x}\chi_{[0,\infty[}(x)$.
I always thought that distributions were set according to a random variable. However, a random variable is not mentioned anywhere above.
My guess is that $f(x):=P(X \in \{x\})$, where $X$ is the random variable that actually has exponential distribution to parameter $\lambda$, rather than the density function $f$.
I need clarity on the terms, and perhaps an explanation on how it all fits together.