Let $X$ be a random variable on $\mathbb{R}$ with known density and distribution function $f_X$ and $F_X$, respectively. $f_X, F_X$ and $F_{X}^{-1}$ are known. Let $M=(m,n)$ be an interval s.t $P(X\in M)>0$.
I need to find $F_{X|X\in M}(x)=P(X\leq x|X \in M)$, $f_{X|X\in M}(x)$ and $F_{X|X\in M}^{-1}(x)$.
My attempt to find solution:
$$F_{X|X\in M}(x)=P(X\leq x|X \in M)\\=\frac{P(X\leq x,X \in M)}{P(X \in M)}\\=\int_{m}^{x}f(y)dy=F(x)-F(m)$$
then,
$$f_{X|X\in M}(x)=f(x)$$
For $F_{X|X\in M}^{-1}(x)$, I would find the inverse function of $(F(x)-F(m))$.
Is my attempt correct? Appreciate your help!
Your start is fine: $$F_{X\mid X\in M}(x)=P(X\leq x\mid X\in M)=\frac{P(X\leq x,X\in M)}{P(X\in M)}$$
But then for some reason you seem to forget the existence of denominator: $$P(X\in M)=F_X(m)-F_X(n)$$
Correct for $x\in(m,n)$ would be the continuation:$$\cdots=\frac{F_X(x)-F_X(m)}{F_X(n)-F_X(m)}$$leading to:$$f_{X\mid X\in M}(x)=\frac{f_X(x)}{F_X(n)-F_X(m)}$$
Further it is obvious that we can complete with: $$f_{X\mid X\in M}(x)=0\text{ if }x\notin(m,n)$$