Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 2 / 11F / a / Solution

For independent random variables, the probability density function of a sum is the convolution of probability densities. Since each density is one on $[0,1]$ and zero elsewhere,This is the length of $[0,1]\cap [u-1,u]$. Therefore the density isThe two triangular areas sum to one, as a probability density function must.