Let $X$ have pdf $$f_X(x)=e^{-x} \qquad \text{ for } x \ge 0$$ and $Y$ have $$f_Y(y)=1 \qquad \text{ for } 0\le y \leq 1$$
$X, Y$ both independent. What is the pdf for $Z=X+Y$?
Using convolution formula I get $$f_Z(z)=\int_{-\infty}^{\infty}f_X(z-y)f_Y(y)dy$$
How do I figure out my limits of integration? Since $f_X(x)$ is from $0\leq x \lt \infty$ so should $f_X(z-y)$ be. That makes want to integrate from $0$ to $z$, which is apparently wrong.