PDF of substraction of uniform

Question

Let X and Y be i.i.d. Unif(0, 1), and let W = X − Y . Find the PDF of W.

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$f(w)=\int_{0}^{1}f_X(x)f_Y(x-w)dx=\int_{0}^{1}f_Y(x-w)dx=?$

Answer 1

\begin{align} f_W(w) = \int_\mathbb{R} f_X(x)\; f_Y(x-w)\;\mathrm{d}x = \int_\mathbb{R} \mathbb{1}_{x\in(0,1)} \mathbb{1}_{x-w\in(0,1)}\mathrm{d}x = \int_{(0,1)\cap(w,1+w)} \mathrm{d}x = (1 -|w|)\mathbb{1}_{|w|<1} \end{align}