Suppose we have two independent uniformly distributed random variables on intervals $X$ on $[a, b]$ and $Y$ on $[c, d]$. What is the pdf of $Z=X-Y$ in terms of $a$, $b$, $c$ and $d$.
As $X$ and $Y$ are independent, $$f_{xy}(x,y)=\frac{1}{ b-a}\times \frac{1}{ d-c}$$ Let $ Z=X-Y $ $$F_{z}(z)=F_{z}(Z<z)=F_{z}(X-Y<z)$$ $$ =\int_{?}^{?} \int_{a?}^{y+z?} f_{xy}(x,y)dxdy$$ How set the bounds of this integral properly? If you can explain. I'd be very glad.