Let $X,Y$ be random variables each with density $f(x)=2xI_{x ∈[0,1]}$ , where $I_{A}$ is the indicator function on the event $A$. What is the density of the random variable $U:=X+Y$?
I start by using convolutions, the density of $U$:
\begin{aligned} f_U (u)=(f*f)(u) &=\int_{-\infty}^{\infty} f(x)f(u-x)\,dx \\ &= \int_{-\infty}^{\infty}4x(u-x)I_{x∈[0,1]}I_{u-x∈[0,1]}\,dx\\ \end{aligned}
However, I am struggling to proceed. I assume the indicators change the limits of integration, is this correct? If so does the expression become $$\int_{0}^{1}4x(u-x)I_{u-x∈[0,1]}\,dx$$ Any ideas on how I deal with the other indicator function?