I understand that the conditional PDF of some RV $X$, given some variable $Y$, is written as $f_{X|Y}(x|y)$. Also, I understand that if I want to describe general features of that function, I am to write it as $f_{X|Y}(\cdot|\cdot)$.
However, what if I want to write the conditional PDF of $X$, given that $X$ falls within a certain range (say a real interval $[a,b]$)? Is the following correct: $f_{X|X}(x|x\in [a.b])?$ Also, how would I write this function if I am describing its general features (e.g. say if I want to write that it is increasing in the length of $[a,b]$, for all $x$ in the support of $X$)? Is the following correct: $f_{X|X}(\cdot|\cdot)?$
Thank you.
The conditional PDF of a random variable $X$ given an event $E$ is usually written as $f_{X|E}(x)$. So in this case it would be $f_{X|X \in [a,b]}(x)$. If I wanted to emphasize that it is a function of $a, b$ and $x$, I might say something like: consider the function $$ g(a,b,x) = f_{X|X \in [a,b]}(x)$$