Consider a context in which $f(X)$ is a probability density function for a given random variable $X$, with domain comprised in $[l,u]$ ($l$ = lower bound, $u$ = upper bound).
Now, consider that someone (e.g., an oracle) gives me additional information regarding the domain of the given p.d.f. $f(X)$. That is, someone says to me that the p.d.f. should be located between $[l',u']$ instead of $[l,u]$ with:
- $l'\geq l$
- $u' \leq u$
This in some way is equivalent to state that the variable $X$ now cannot assume values in the interval $[l,l']$ and/or in the interval $[u',u]$ (as suggested by the rough information I had before this oracle spoke).
How does this new information affect the p.d.f. $f(X)$? Is there a way of "cutting away" part of the distribution so as to reshape it in the new interval?
EDIT: I would like now to condition on the fact that $X$ can only take values on $[l',u']$.
Thank you so much for your help.
To condition on the knowledge that $X$ is in $A$, where $A$ is such that $P[X\in A]\ne0$, replace the density $f_X$ of $X$ by the density $f_{X\mid A}$ defined as $$ f_{X\mid A}(x)=\frac{f_X(x)\mathbf 1_A(x)}{P[X\in A]}. $$ That is, $f_{X\mid A}(x)=0$ if $x$ is not in $A$ and, if $x$ is in $A$, $$ f_{X\mid A}(x)=\frac{f_X(x)}{P[X\in A]}. $$ The case you describe is when $A=[\ell',u']$.