Conditional probability the of Sum of two independent random variables

Question

Let $X$ and $Y$ be two independent random variables such that $X>a$ and $a<X+Y<b$. Please how do I simplify further, the conditional probability $\Pr(a <X+Y <b \Big| X>a) $ $~$ ? I am guessing that one of the final terms will involve convolution of the sum $X+Y$, but I don't know how to go about it. Thanks

Answer 1

It is $\frac {\int_{(a, \infty)} [F_Y((b-x)-)-F_Y(a-x)] dF_X(x)} {\int_{(a, \infty)} dF_X(x)}$.

[ Write $\{a<X+Y<b, X>a\}$ as $\{a-X<Y<b-X, X>a\}$, conidition on $X$ and then integrate].