Let $X$ be a random variable with a CDF. $a\ ,\ b\ \in \ R$ such that $a<b$ .
Let $G$ be a function that $G\ :\ R\rightarrow R$
$$G(x)\ =\ P(X\leq x\mid a<X<b)$$
I want to write function $G$ in terms of $F_{x}$ , $a, b$ and determine whether $G$ is a cumulative distribution or not.
I started with the conditional probability and wrote $$P(X\leq x\mid a<X<b)\ =\ \frac{P\ (\ a<\ X\ \leq \ b\ \bigcap \ X\ \leq \ x)\ }{P\ (\ a<\ X\ \leq \ b\ \ )} $$
Then, I know that
$$P\ (\ a\ <\ X\ \leq \ b\ )\ =\ F_{x}(b)\ -\ F_{x}(a)$$
$$P\ (\ X\ \leq \ x)\ =\ F_{x}(x)$$
But I don't know how to go from there. Can someone help?