Solution
= Solution
Write $X_i=\theta+U_i$, where the $U_i$ are independent $\operatorname{Uniform}(0,1)$ variables. Convolution gives the <sum of two independent uniform variables>:
$$
f_{X_1+X_2}(s)=
\begin{cases}
s-2\theta,&2\theta\leq s\leq2\theta+1,\\
2\theta+2-s,&2\theta+1\leq s\leq2\theta+2,\\
0,&\text{otherwise}.
\end{cases}
$$
Its plot is a triangle with vertices
$$
\boxed{(2\theta,0),
\qquad(2\theta+1,1),
\qquad(2\theta+2,0).}
$$