Distribution of the Sum of Two Independent Uniforms

Let $X$ and $Y$ be independent $\text{Uniform}(0, 1)$ random variables. Define $S = X + Y$.

(a) Using the convolution formula $f_S(s) = \int_{-\infty}^{\infty} f_X(t)\, f_Y(s - t)\, dt$, derive the PDF of $S$ for all $s \in \mathbb{R}$.

(b) Sketch the PDF and identify the distribution by name.

(c) Compute $P(S > 1.5)$.