CDF of sum of 3 dependent random variables Given three dependent random variables, $S_1, S_2$ and $S_3$, such that $0 &lt; S_1, S_2, S_3 &lt; \infty$ and assuming known their joint PDF $f_{S_1,S_2,S_3}(s_1,s_2,s_3)$ I would like to find the CDF of their sum, $E := S_1 + S_2 + S_3$. Basically, I want to find $F_E (e) = P(E \leq e)$.

I know that

$$F_E(e)=P(E \leq e)=P(S_1+S_2+S_3\leq e)=∫^?_0∫^?_0∫^{e−s_1-s_2}_{0}f_{S_1,S_2,S_3}(s_1,s_2,s_3)dS_3dS_2 dS_1$$
but I'm unsure of what is the upper limit in the other two
integrals (those on $S_1$ and $S_2$): is it ∞ or $e$?
Hint:

$$\iint\limits_{x+y\leq z}f_{X,Y}(x,y)dxdy=\int\limits_{-\infty}^{\infty}\left(\int\limits_{-\infty}^{z-x}f_{X,Y}(x,y)dy\right)dx$$

$$\iiint\limits_{S_{1}+S_{2}+S_{3}\leq e}f_{S_{1},S_{2},S_{3}}(s_{1},s_{2},s_{3})ds_{1}ds_{2}ds_{3} = \dots$$
I think we can solve it in this way:

$F_{E}(e)=P(E&lt;=e)=\int_{-\infty}^{+\infty}ds_{1} \int_{-\infty}^{E-s_{1}}ds_{2} \int_{-\infty}^{E-s_{1}-s_{2}}f_{S_{1},S_{2},S_{3}}(s_{1},s_{2},s_{3})ds_{3}$

but I'm not sure.
I like to know the solution too.