If $X \sim U(a_1, b_1)$ and $Y \sim U(a_2, b_2)$ where $a_1<a_2<b_2<b_1$ (The first one has a higher standard deviation) how do I compute the density of $Z=X+Y$ using the convolution? I think Im getting confused while doing it. Here is what I have:
Denoting $f_x$ and $f_y$ the pdfs, th convolution is
$(f_x * f_y)(x)=\int_{-\infty}^\infty f_x(x-t)f_y(t)dt$
I divide it into different parts
$\int_{-\infty}^{a_1} f_x(x-t)f_y(t)dt+\int_{a_1}^{a_2} f_x(x-t)f_y(t)dt+\int_{a_2}^{b_1} f_x(x-t)f_y(t)dt+\int_{b_1}^{b_2} f_x(x-t)f_y(t)dt+\int_{b_2}^{\infty} f_x(x-t)f_y(t)dt$
Now im having troubles identifying how to compute the intrgals that are not zero, since the values of $f_X(t)$ are not the same as the ones for $f_X(z-t)$
Any help is appreciated, thanks