We have found the density of the sum of two independent U(0,1) variables. Use that result to find the density of the sum of three independent U(0,1) variables.
(You will need to treat intervals (0,1), (1,2) and (2,3) separately, although an appeal to symmetry can reduce the work. Give yourself a gold star if you get the right answer first go - get the limits of the integrals correct).
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The density of the sum of independent variables are discussed. The solution is detailed and well presented.