Say $f_X(x), f_Y(y)$ are two random independent PDF's with domains $D_X$ and $D_Y$. Let $Z = X + Y$. I know $f_Z(z) = f_X \star f_Y(z)$ where $\star$ means convolve. How do I determine how to integrate this given only the domains of the random variables?
I am trying to solve this for $X - U(0,1)$ and $f_Y(y) = a^{-y}$ but I realized I don't understand how to choose the bounds in general.
For my example I would think to do $\int_0^1a^{-(z-x)}dx$ but I don't understand how $z$ would have anything to do with this.
Anyone have any ideas?
$f_X(x)=1_{(0,1)}(x)$ and I'll assume $f_Y(y)=a^{-y}\cdot 1_{(0,\infty)}(y)$ and moreover $a=e$. Then for $z\ge 0$, $$f_Z(z)=\int_{-\infty}^{\infty} f_X(x)f_Y(z-x)\,dx=\int_0^1 a^{-(z-x)}\cdot 1_{(0,\infty)}(z-x)\,dx$$ $$=\int_0^{\min(z,1)} a^{-(z-x)}\,dx=e^{-z}\int_0^{\min(z,1)}e^{x}\,dx$$ $$=e^{-z}(e^{\min(z,1)}-1)=\begin{cases}1-e^{-z}&\text{if }z<1,\\ e^{-z}(e-1)& \text{if }z\ge 1.\end{cases}$$ So maybe the point that you were missing is how to work with the domains, like $$\int_{-\infty}^\infty 1_{(a,b)}(x)1_{(c,d)}(x)f(x)\,dx=\int_{-\infty}^\infty 1_{(a,b)\cap(c,d)}(x)f(x)\,dx=\int_{\max(a,c)}^{\min(b,d)}f(x)\,dx$$ if $\max(a,c)<\min(b,d)$.