Let $Y,X$ be positive independent continuous random variables with densities $f,g$, and $a<b$ constants. For example $f,g$ could be exponential or gamma densities. Let $Z=X+Y$ be the sum of $Y,X$ so that the distribution of $Z$ is the convolution of $X,Y$; call its density $h$. Consider the joint probability of the events $Z \le b$, and $a < X$.
$$P(Z \le b , a < X)$$
Is there a way to express this joint probability in terms of its densities $f,g,h$ or the respective distribution functions $F,G, H$? For example
$$P(Z \le b , a < X) = P(Z \le b | a < X) P(a < X)$$
with $P(a < X)=1-F(a)$ but I am not sure about how to simplify the first factor. Somebody told me $(1-F(a))H(b-a)$ may be one way, but I could not show it.
$P(Z< b, a<X) = P(a<X<Y-b) = \int_{x,y} f(x)g(y)\mathbb 1_{a<x<y-b} dxdy = \int_b^\infty g(y) (\int_{a}^{y-b} f(x)dx)dy = \int_{a+b}^\infty g(y)(F(y-b) - F(a))dy = \int_{a}^\infty g(t+b)(F(t)-F(a))dt$.
I doubt anything would come out of $(1−F())H(−)$ because it seems to assume that $X$ and $Z$ are independent, which is not true.