For two density functions: Suppose again that $Z = X + Y$.
Find $f_Z(z)$ if
$$f_X(x) = f_Y(x) = \begin{cases} x/2, & \text{if $0\lt x\lt 2$} \\ 0, & \text{otherwise} \end{cases}$$
I understand this is a convolution $f_X \star f_Y$ but I don't understand how to obtain the limits of integration since they are not $-\infty$ to $+\infty$.
$X = x \\ Y = Z - x$
$f_Z(x) = \int {f_X(x)f_Y(z-x)dx}$
$0 \lt x \lt 2 \\ 0 \lt z-x \lt 2$
so:
$x \leq z \\ z-2 \leq x$
and now I'm stuck.
Lower bounds on $X$ are both $0$ and $Z-2$, so $X\gt\max\{0,Z-2\}$.
Upper bounds on $X$ are both $2$ and $Z$, so $X\lt\min\{2,Z\}$.
So the integral is
$$f_Z(z) = \int_{x=\max\{0,z-2\}}^{\min\{2,z\}} f_X(x)f_Y(z-x)\;dx.$$
Or, to split into two cases:
For $0\lt z\lt 2$,
$$f_Z(z) = \int_{x=0}^{z} f_X(x)f_Y(z-x)\;dx.$$
For $2\leq z\lt 4$,
$$f_Z(z) = \int_{x=z-2}^{2} f_X(x)f_Y(z-x)\;dx.$$